Film-makers are not, for the most part, paragons of cool, but Sofia Coppola is different – a director whose pristine aesthetic extends beyond the screen into what her close friend the fashion designer Marc Jacobs admiringly terms an “art of living”. Stephen Dorff, star of her 2010 film Somewhere, deems her coolness contagious: “When she casts me, everyone thinks I’m cool again.” And down in the more lowly ranks of film criticism, standard-issue scepticism becomes fannish enthusiasm when I mention I’m about to interview her: “Oh God, I just want her life,” one friend gushes.

Six feature films into her career, Coppola remains unique in a film industry that, in particular, rarely makes celebrities of female directors. Born into the Hollywood firmament – the daughter of era-owning auteur Francis Ford Coppola – she came to film-making via acting, modelling and fashion design. She’s married to a French rock star, Phoenix frontman Thomas Mars, and they divide their time between Manhattan’s West Village and Paris.

Yet Coppola’s coolness remains principally defined by her work. The gorgeously designed but sincerely dreamy sense of melancholy that runs through her films – from the faintly autobiographical lonelyhearts valentine Lost in Translation to the swirling historical reverie of Marie Antoinette or the spiky, morally ruthless youthquake of The Bling Ring – makes them seem more alike than they are. They seem to start from her own impeccable image and then build on it: Scarlett Johansson’s pensive Tokyo drifter from Lost in Translation is sealed in our imaginations, accurately or otherwise, as Coppola’s alter ego.

“Oh God, really? I’m cool?” Coppola says shyly. “It’s just funny if people think that,” she murmurs. “I was watching some clips of myself on the red carpet from last night and I feel, like, so dorky on them. I’m really nerdy. And I live with a 10-year-old who thinks I’m very uncool.”

That 10-year-old – Romy, the elder of the two daughters Coppola has with Mars – is several thousand miles away from the vast, bay-view suite in Cannes’s Martinez hotel: a plush pile-on of old-school Riviera glamour, preserved in palest amber from the days of Hitchcock’s To Catch a Thief. Luscious bouquets of cabbage-sized, blush-coloured peonies (a gift from Chanel) adorn the room; Coppola, barefoot in wide-legged jeans and an origami-crisp white shirt, black spectacles balanced a few degrees askew on her nose, is giving nerdery a makeover.

Coppola’s sixth film, The Beguiled, has just premiered here at the Cannes festival. When we meet in the early evening, she’s exhausted and halfway through packing, restless to get home to her family in New York the next morning. “I have to get back to real life,” she says. “Everything is so heightened here.”

She’s not planning to return for the Cannes closing ceremony – where, as it turns out, she ends up winning the best director prize, only the second woman in the festival’s 70-year history to do so. At this point, Coppola’s only concerned with audiences liking the film, a sad, sensual, wickedly funny slice of southern gothic based on the civil war-set novel by Thomas P Cullinan – and about a million miles removed from Don Siegel’s lurid, sweatily macho 1971 adaptation of the same text.

She has reason to be nervous, and not just because the film’s taut thriller trappings and sly streak of high camp mark it as something of a departure for her: it’s the first film she’s presented in competition at Cannes since 2006’s Marie Antoinette, the proud teen-girl perspective and brazen anachronisms of which earned her a pelting of literal boos from the festival’s dour critical patriarchy. “It’s always a lot of pressure showing anything you’ve made, and I knew that film was more obnoxious,” she admits. “But it’s still nerve-racking coming back.”

Now 46, Coppola has both experience and critical revisionism on her side. A decade ago, after her Oscar win for Lost in Translation, many critics had knives out for a film-maker who’s always had to fight accusations of silver-spoon privilege – from a male-dominated industry guilty sometimes of writing off the “girlier” aspects of Coppola’s film-making as mere frippery. Years have been kind to Marie Antoinette, however: its images have endured, its critical standing has strengthened. “People often come up to me and say how much they enjoyed it, so I feel like it’s settled in and found a place,” she says. “It’s gotten more of a life now than when it came out.”

Cannes laurels notwithstanding, critics are also fiercely taking sides over The Beguiled. Following its US release last month, controversy has brewed around allegations of Coppola “whitewashing” Cullinan’s text, following her decision to remove the supporting character of a black female slave for her version, and to cast Kirsten Dunst in a role presented as biracial in the novel. The film’s beautifully designed period trappings “evince a commitment to authenticity, [but] no interest in the people on whose backs that lavish lifestyle was built,” complained Slate; in subsequent interviews, Coppola has calmly defended her decision “not [to] brush over such an important topic in a light way,” adding: “Young girls watch my films and this was not the depiction of an African American character I would want to show them.”

What some might see as deflection strikes me as a kind of humility, a director acknowledging the specificity of her own worldview. It’s not the first time Coppola’s films, many of which centre on the frustrations of white characters blessed with enviable wealth and/or celebrity, have been accused of betraying the social and cultural privileges of her own upbringing. Such a concentration shouldn’t be mistaken for celebration, however, and the same goes for The Beguiled: it’s a film tacitly haunted by suggestions of abuse and oppression.

The Beguiled stars Coppola favourite Kirsten Dunst, left, and Nicole Kidman, centre. Photograph: Ben Rothstein/Focus Features

Gender politics are foremost. The story of an injured Union soldier (Colin Farrell) grudgingly sheltered in a Mississippi ladies’ seminary overseen by Nicole Kidman’s flinty schoolmarm – his presence causing social and sexual discord among its young inhabitants – the film apportions guilt and vengeance far more ambiguously than Siegel’s version, which fashioned it as a grand guignol fable of male victimhood. In Coppola’s, each immaculate frame is suffused with female desire, pulling the audience’s sympathies from character to character, resting eventually with the quiet, lonely yearning of Kirsten Dunst’s junior schoolmistress.

For Coppola, Siegel’s film wasn’t a cherished or sacred text. “I wasn’t very familiar with it, to be honest, and it’s not very well known in the States except among cinephiles,” she says. “But Anne Ross, my production designer and an old friend, told me that I had to see it, that I should remake it. It became kind of a joke to us. But I watched it and it just stayed in my head – because it was so weird. I really loved the whole premise. I just thought a story about this enemy soldier coming into this southern girls’ environment was so interesting, and that I’d love to kind of flip it: to tell the same story but from the women’s point of view, what it must have been like for them. Not a remake but a reinterpretation. ‘Remake’ is like a bad word in our family. I would never.”

From its female gaze to its dusty-rose palette, the film is certainly feminised – but did Coppola intend it as a feminist work, as some critics claim it is? “I don’t love that labelling myself,” she says. “I’m happy if other people see it that way, but I just see it as having a female perspective, which isn’t always the same thing to me.”

Coppola prefers not to think schematically about such things. She has admitted to being unfamiliar with the Bechdel test, the frequently cited academic measure of the extent to which female characters talk to each other in the space of a screenplay – a test The Beguiled nonetheless passes with flying colours. “It’s a story about strong women, and I wanted to look at these characters in more depth, where in the 1971 film, they’re more kind of crazy, or just caricatures. You don’t really know as much about them.”

For the New York-born, California-reared Coppola, it was also a chance to explore a different part of America. “The south is always very exotic to me: I wanted the film to represent an exaggerated version of all the ways women were traditionally raised there just to be lovely and cater to men – the manners of that whole world, and how they change when the men go away.”

Though it’s set in the 19th century, she wanted the setting and its gender tensions to retain a contemporary resonance. “I thought of seeing Gone With the Wind as a kid, and then tried to make something that’s naturalistic – to be relatable even though it’s another era. Even with the costumes and hair, we tried to refer to the period but make it so you could connect.” In this regard, Coppola’s fashion sensibilities came into play: Stacey Battat’s costumes appear half dredged from the attic of a plantation mansion, and half filched from the Chanel catwalk.

Coppola delights in such prettiness, and not just for its own sake – The Beguiled, she felt, was a necessary corrective to the harsher, harder Los Angeles glare of The Bling Ring, a caustic drama based on the true story of a gang of bored, wealthy Beverly Hills teens who take to celebrity house burglary. “That movie was so ugly to me, so I really wanted to follow it with something beautiful, something poetic after all that trash culture,” she says. “Every movie is affected by the one before it. This one actually reminded me of Virgin Suicides, that kind of pastel world – I wanted the story to be very feminine and non-threatening, until a man comes into it.”

In an industry where a feminine point of view continues to be devalued – where harder-edged genre directors like Kathryn Bigelow are praised for supposedly proving themselves on a man’s turf – the aesthetic delicacy of Coppola’s cinema, with its undisguised interest in fashion, floral motifs and tender female beauty, is its own kind of defiant statement. Eighteen years into her directorial career, Coppola has weathered any number of sexist jibes against her work – dismissing her aesthetic as decorative or insubstantial in terms that comparable stylists such as Wes Anderson hear less often – and remains unfazed.

In the US, the film’s marketers have explicitly pitched the film to a young female market, with a Twitter campaign heavy on hashtags like #Squad, #VengefulBitches and #HBIC (Head Bitch In Charge, if you’re wondering). It’s not an intuitive route, but Coppola is wryly accepting of the vagaries of marketing. (“Does this pose say, ‘Come and see my film’?” she quips to our photographer three weeks after Cannes, when we meet in London for the Observer’s photoshoot.) Whatever form it takes on social media – a playground Coppola personally avoids – she is glad to play up the “girlier” aspects of her work.

“I just feel like I have a feminine point of view and I’m happy to put that out there. We certainly have enough masculine ones,” she says. “I never felt like I had to fit into the majority view. Maybe growing up with so many strong men around me meant I felt, I don’t know, closely connected to being feminine. I mean in my first movie I felt like making something for teenage girls. I looked at the movies they made for teenage girls and thought: why can’t they have beautiful photography? Why shouldn’t we treat that audience with respect? That was something I missed when I was that age: I wished the movies weren’t so condescending. So I guess I’ve always just made the films that I’d have wanted to see.”

This is something of a luxury, as Coppola doesn’t mind admitting. As the daughter of the industry titan who made Apocalypse Now and the Godfather films – and whose production company, Zoetrope, has backed all her features to date – she hasn’t exactly met the usual challenges faced by other women trying to break Hollywood’s glass ceiling. But that’s not to say she hasn’t felt the sting of gender discrimination: being the daughter of a major name has as many cons as it does pros.

“When I first started, people would say things like, ‘Oh, your casting was really well done. Did your father or your husband help you?’ [At the time, Coppola was married to director Spike Jonze.] And that was really insulting, obviously. You wouldn’t say that to a male director.” Does it annoy her still to be referred to as Francis Ford Coppola’s daughter, then? “It very much depends on the way it’s said. I’m proud to be his daughter. I learned to have balls from him, and integrity. But I have a body of work now, and it has its own identity. He’s a great master, but I’m happy to carve out my own way of working.”

Coppola with her husband, Phoenix frontman Thomas Mars. Photograph: Alessandra Benedetti/Corbis via Getty Images

Coppola’s brushes with industry sexism are one reason she prefers to keep working in the independent sphere, where gender disparity is less pronounced. “When it comes to bigger-budget studio movies, there’s definitely a bias – it’s harder for them to hire a woman,” she says. At the film’s Cannes press conference the day before, Nicole Kidman cited a statistic stating that only 2% of Hollywood studio projects are female directed; Coppola admits the figure is even lower than she thought. Her own lone brush with studio film-making came to nothing: attached in 2015 to direct a live-action version of The Little Mermaid for Disney, she backed out over the old industry chestnut of “creative differences”.

“My experience was that when there’s a big budget there’s less creative freedom,” she says of the experience. “There’s so many cooks in the kitchen because it’s more about business than art. And I was excited because I thought oh, you know, I could try to change this or that, and it wasn’t that easy. So I was so glad to then work on The Beguiled, which was contained and I could really control every aspect of it. And you can do that on a low budget. Even then, I’m always surprised at how hard it is to get another film off the ground. It always feels like your first movie all over again.”

Coppola is dubious about the state of Hollywood cinema, which she sees as a more risk-averse realm than it was in her father’s 1970s heyday. “It’s such a conservative time,” she sighs. “Studios just don’t want to make weird movies. Kirsten [Dunst] and I were talking last night about Harold and Maude: imagine pitching a love story about an 80-year-old woman and an 18-year-old guy today. Luckily I can raise finance and do things separately. But I do feel like the movies that are being made by studios are pretty...” she trails off, and briefly makes a face. “It’s not a great time.”

She means “conservative” in a creative sense, though I wonder if there’s a political edge to her weariness: when I bring up the T-word that is all but unavoidable in any conversation with an American these days, her shoulders visibly sag. “It does feel strange being in New York now, with all of that going on and then having Trump Tower just looming here over us,” she says darkly. After several years in Paris, she and Mars made New York their home base in 2010; despite recent political developments, they’re contentedly ensconced there with Romy and seven-year‑old Cosima.

Scarlett Johansson and Bill Murray in Coppola’s 2003 romantic comedy-drama Lost in Translation. Photograph: Focus/Everett/Rex Features

“We live in the West Village, which feels like its own different world, the kids go to school there, and it’s a nice place to be,” she says. “Tom’s band is still based in Paris, and there’s our French family, so we’ll go there in the summer. So we get a bit of both.”

Coppola’s guard goes up slightly when the subject turns to her family: she keeps a determinedly low public profile, insistent that her daughters’ lives go largely uninterrupted by her career and her travels. As something of a Hollywood brat herself – just a teenager when she took cruel critical brickbats for playing a role in The Godfather III at her father’s request – is her low-exposure parenting the product of life experience? “I don’t want them ever to be jaded,” she says. “I never saw the point of taking little kids to movie premieres and stuff. I just want them to have a childhood.”

Watching movies, on the other hand, remains a treasured family activity. “The girls love them, but they have really bad taste!” she says, eyes rolling affectionately. “I feel like I was exposed to all these great movies at their age, but if there’s anything classic, they’re like no, we’re not watching an old movie. So that’s funny.” It’s reassuring to know that Sofia Coppola, like any other parent out there, has also been enslaved to Frozen.

Things change, of course: Coppola herself admits to being bored by many of the canon classics that were on rotation in her parents’ household growing up, until experiencing a breakthrough, aged 13, with Jean-Luc Godard’s Breathless. And those tastes, she says, are transient: “I feel my films are a snapshot of that moment and who I was, where I was, at that time. I don’t revisit them much.” Has she considered what The Beguiled reveals about her now? She frowns. “I think I’m still far too close to it to say – ask me in a few years’ time,” she replies. “But what’s different about this one for me is that I’ve now been the age of each of the major characters, so I have more insight into each of them and their level of maturity. So perhaps it’s more than one snapshot at a time now.”

As such, she has little sense of where her tastes might drift to next – though she has one dream collaborator on her wishlist you wouldn’t necessarily guess. “I would really love to direct Eddie Murphy,” she says, a slight twinkle disrupting her thoughtful expression. “I’ve always been such a fan of his, and I wish he would do something interesting.” I’m intrigued to find out where the worlds of Sofia Coppola and Eddie Murphy might collide, I say.

“So am I,” she laughs. “I’ll try to figure it out.”

The Beguiled is out on 14 July