Cara Delevingne likes to think of herself as an accidental supermodel, and certainly she is unlike any other catwalk star I have ever met. She arrives for our interview at the Chiltern Firehouse - but of course - on time and with an entourage that consists of a tiny pet rabbit called Cecil, which she adopted after having to accessorise with him on a shoot. "Peeing! Peeing!" she screams, as the rabbit relieves itself next to my tape recorder. "I'm a good mother," she says, getting on her hands and knees and clearing up the mess her 'child' has left on the floor.

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She wants to do the interview in bed, and who am I to pass up a pseudo-sapphic encounter with a woman who has been linked with everyone from Harry Styles to the actress Michelle Rodriguez? The reason for the 21-year-old's eagerness to jump under the covers with me is rather more prosaic, however. She has just got back from her sister Poppy's wedding party, a five-day bash in Marrakesh, and she is exhausted. Cara has a reputation as something of a party girl - she is frequently snapped out on the town with friends such as Rihanna, Miley Cyrus and fellow model Suki Waterhouse - but even she struggled to keep up with the celebrations. "I spent all day yesterday eating crap food and watching bad TV," she says, tucking into a meal that consists of pancakes, some eggs Benedict, and a portion of chips. "It was f***ing great," she says dreamily. "Sorry, can I swear?

Delevingne is thoughtful, chatty and engaging; she wears her heart on her Chanel sleeve (today she is wearing a Chanel top, some J Brand jeans and a pair of Union Jack socks; she says that "the best thing I get sent now is socks as I used to always have bags of odd ones"). We are here - ostensibly - to talk about the range of handbags she has designed for Mulberry (which start from £795), but she can't help veering off topic, much to the chagrin of the two PRs who have been dispatched to sit in the room with us.

But it is her openness that has won her an army of fans: 1.68 million on Twitter and an astonishing 5.6 million on Instagram. She has been described as a sort of anti-model, the very opposite of the notoriously silent Kate Moss; Delevingne is the first fashion face to really embrace social media. "I think it's quite strange [that nobody had done that before her]," she says. "I think it's nice to break down that barrier, that models are seen and not heard." It does not seem to have done her career any harm, going from Asos to DKNY and Burberry via three British Vogue covers and one Model of the Year award in only four years.

She is not afraid to post a picture of herself and Rodriguez hooked up to hers-and-hers drips after a weekend of partying, and when I mention her honesty she says, "Yeah, which is apparently quite bad sometimes. I mean, Kate [Moss] is incredibly open, but she chooses who she is open to. I'm just open, generally," she says, laughing. "But I feel that's when you really connect with someone." When I ask her about her sexuality one of the PRs suggests politely that I have veered too far off topic, but Delevingne doesn't think so. "On that topic, I think..." She pauses for a while, gathers her thoughts behind her big bushy eyebrows, which also have their own Twitter feed. "What do I think? I think people shouldn't be scared of that. I'm young, I'm having fun, I don't want to pretend to be something I'm not." She shrugs. "So I don't really care on that matter. People can say what they want, but I'm having a good time. I know what people are doing who are my age; I just think it would be a lie to pretend that I'm not having a good time."

Oversized cream washed silk stripe tuxedo shirt, £850, Cara Delevingne bag in khaki camouflage haircalf, £2,200, both Mulberry (020-7491 3900)

She stops short of a full outing but she tells me that she is more comfortable in the company of women. "The funny thing is, I always used to have more guy friends. At school, I was a tomboy and it would be me and all my guy friends. But now... I don't know. It's kind of changed quite a bit. I still have my old friends from school, but I think..." She lets out a big sigh. "I don't meet men now who just want to be my friend. It doesn't really happen that much. They're just shallow like that. Unless it's one of my girlfriend's boyfriends, most guys are not like, 'Oh, we should just be mates.' I think it's harder to become friends with guys," she says, looking a little sad, "because guys just want to have sex with you."

Cara Jocelyn Delevingne was born 21 years ago to Pandora, a socialite and former heroin addict, and Charles, a property developer whose grandfather was a viscount. Her maternal grandfather is Sir Jocelyn Stevens, the former newspaper executive and publisher of Queen magazine, while her maternal grandmother was a lady-in-waiting to Princess Margaret. Cara has two elder sisters - Poppy, also a model, and Chloe, a biomedical science graduate who has just given birth to her first child - and when she was growing up she wanted to be either a musician or an actress. She had her first taste of modelling when she was 10, when she worked on a Bruce Weber shoot for Italian Vogue, but Delevingne only remembers it now as all being a bit "odd".

At 17, while at the boarding school Bedales, she came very close to getting the part of Alice in Tim Burton's Alice in Wonderland, but then modelling happened (Sarah Doukas, the head of Storm model agency, spotted Delevingne because her own daughter was at the same school) and acting was put on the back burner. "I never really thought about modelling," Delevingne says now, "it wasn't something I ever wanted to do. I used to always be so angry about modelling. I was always like, 'F***, I'm never going to able to act because I'm a model.' " But she now sees that her fashion career has given her fantastic opportunities. The Mulberry bags, for a start, which have been like a "literal dream come true. The first proper handbag I had was the Alexa [named after Alexa Chung] and I wore it every day until it broke. And I remember actually meeting Alexa for the first time and being, like, 'Ohmygod she's got a fricking handbag!' When they asked me I couldn't believe it, I thought they were lying."

She has been involved closely with every stage of the design - the Cara can be worn as both backpack and handbag - because "if I put my name to something, and I am criticised for it, I want to know that I've at least had my full stamp on it." Does she get asked to put her name to a lot? "Not much," she sighs. "I'm quite surprised that nobody has asked me to do my own line of tweezers. I totally would love to do that. Or, like, mascara. Cara's Mascara! Ahahahaha."

Anyway, I'm not sure she is going to have time for all of that because the acting career is now taking off. Earlier this month she starred with Sylvia Syms in Timeless, a drama for Sky Arts written by Tim Frith of Calendar Girls fame. She was really very good, like a "young Audrey Hepburn" according to no less an authority than Syms herself. Next she stars alongside Kate Beckinsale and Daniel Brühl in Michael Winterbottom's The Face of an Angel, which is very loosely based on the circus surrounding the Meredith Kercher murder trial (Delevingne plays Brühl's love interest in the movie and not, as previously reported, the Amanda Knox character).

Mini Cara Delevingne bag in indigo quilted nappa leather, £995, Mulberry (020-7491 3900)

Then there is the big-screen adaptation of Martin Amis's London Fields, in which she stars as Kath Talent, the wife of Jim Sturgess's Keith Talent. She also got to work alongside Amber Heard and Billy Bob Thornton, and "I got so scared, because they are the most incredible actors. But fear and nervousness are something that can hold you back quite majorly. I just wanted to be inside their head, I just wanted to learn." She was terrified doing her first scene with Thornton "and he saw that. But he said, 'As an actor I can spot straight away when someone can act, and you can act.' " She blushes at the memory. "He told me you can't teach a bad actor to act, but you can make a good actor worse. So he was like, 'Don't feel you have to learn.' "

She does feel guilty that all her friends are struggling at drama school while things are taking off for her, but "I always feel that life can teach you how to act. I'm always looking at life through other people's eyes. By feeling empathy. And I do feel that I am constantly learning." She says that modelling can interfere with acting "because it makes you so self-aware, and I have definitely had to take a minute [to get used to that]."

Mini Cara Delevingne bag in indigo quilted nappa leather, £995, Mulberry (020-7491 3900)

We talk a bit about the madness of the past four years. "When people described things to me as a whirlwind, I never really knew what it meant," she says. "But then you get inside the tornado and you know exactly what it is. It's madness. You get so caught up in so many things that it can be difficult to find the time to say, 'Am I actually OK with this? Am I happy?' " Does she ever dream of a different story, in which she is like every other 21-year-old out there?

"Ahhhh," she says, a look of sudden intense thoughtfulness on her face. "You know, I love putting myself in other people's shoes, but I can never really experience it, which I find really sad. I love talking to my friends at uni and seeing what they are doing. They're just finishing their dissertations and I kind of wish I could live their life for a second. I wish my school days could have dragged on a little longer, or that I could go back and do it later in life. I always used to get really depressed…" She falls quiet for a moment. "Well, not really depressed, but really upset if everything was getting a bit crazy, because I can never take this all away for a minute. I can't turn around, you know? I've got to keep on going. If there's a moment where I just want it all to disappear and go back… well, I can't. And that used to terrify me."

I really like Cara Delevingne. I can imagine taking her down the pub and having a really good time. She is very easy to talk to, very relaxed, incredibly gregarious. How old does she feel, I wonder. "Mmm," she says. "I'd say 15 going on 32!" She lets out a very long laugh. "It's very confusing. But I know how lucky I am." Then Cecil the rabbit relieves himself again, and Cara runs off looking for a towel.

Wool cashmere rib wrap in grey marl, £450, Mulberry (020-7491 3900); lumière mango yellow low-waist boy shorts, £90, Eres ; large Cara Delevingne bag in Delevingne green quilted nappa leather, £1,400, Mulberry (020-7491 3900)

With thanks to Storm Models and Mulberry

Fashion Director: Daniela Agnelli

Photographer: Sonia Sieff

Model: Cara Delevingne at Storm Models

Photographer's assistant: Vivian Daval

Hair: Raphael Salley at Streeters, using Schwarzkopf

Make-up: Andrew Gallimore at CLM Hair & Make-up, using Dior Transat Summer Look and Dior Capture Totale Dreamskin

Nails: Tinu Bello using Essie in school boy blazer and Creme de la Mer

Fashion assistant: Rachel Hannaway

Shot at Chiltern Firehouse