On a wet day in Reykjavik, the rain battering the fishing boats, the tourist shops and the young male artists with their improbable moustaches, Iceland's minister of industry, energy and tourism is explaining to me that the country needs to be "more badass" about the gender pay gap. The minister is Katrin Juliusdottir, a warm, attractive woman in her mid-30s, pregnant with twins. As she speaks, a hint of frustration enters her voice. Icelandic legislation supposedly guarantees equal pay for equal work, as in the UK, "so why don't we have more penalties?" she says. "Maybe we need to be even more badass when it comes to people breaking the rules."

We are sitting in Katrin's office (all Icelanders go by their first names), in an anonymous building a few hundred yards from Reykjavik harbour, and she is talking about women's rights with no-nonsense passion. Yes, of course she is a feminist; no, she wasn't in the country for the last major women's march, otherwise she would certainly have attended; yes, it's good that the current Icelandic cabinet has four women and six men, but it's not enough. She would like to see it reach the perfect 50/50. (The current UK cabinet is 86% male.) Following the disastrous collapse of the Icelandic banks in 2008, she says, the country "wants balance in our lives, and a big part of that is the balance between men and women."

Some would say this balance already exists in Iceland – that the country is, in fact, the closest the world has to a feminist paradise. For the last two years it has topped the World Economic Forum's report on equality between the sexes, and last month Newsweek named it the best place in the world for women. The Newsweek survey looked at health, education, economics, politics and justice, and found that in all areas, and the last one in particular, Iceland is about as good as it gets. The prime minister, Jóhanna Sigurdardóttir, tells me via email that she's proud of the survey's outcome, "and not only for women, [but because] we know that gender equality is one of the best indicators for the overall quality of societies."

Through the cold mist on Laugavegur, Reykjavik's main drag, I ask Icelandic women what they think. Gudrun, 72, peers shyly from her voluminous hood and says while she loves Iceland – its cleanliness, beauty, the proximity of hot springs, volcanoes, glaciers – it can't possibly be the best place in the world for women "because we don't get the same salary as men".

Awareness of this issue is running high because of a campaign by the commercial and office workers' trade union, VR. To emphasise and redress the fact that Icelandic women are paid, on average, 10% less than their male colleagues, it last month set up a temporary discount of exactly that amount for all female customers at a range of major shops. Berglind, a young shop worker with a metal bar through her septum, tells me she'd like to see classes for teenage girls on how to negotiate hard with bosses.

Erla, 37, a lawyer, swaddled in a thick, red mac, says that as an Icelandic woman you can always count on the support of your sisters, and it was in this spirit she attended the Women Strike Back march last year, a protest against the pay gap and sexual violence. "I don't think I suffer from unfair pay now," she says, "but I have done, and I felt I needed to support women, because we didn't come this far as a society by accident. It was because people went out and worked for women's rights."

To an outsider's eye, the power of Iceland's feminist movement is astonishing. The country was the poorest in Europe before the second world war, but saw a boom afterwards, and by the late 1960s a whole generation of educated women was coming of age and feeling angry about wage inequality. Those who remained in the home felt similarly undervalued. In 1975, a one-day women's strike was proposed by radical feminist group the Red Stockings. The concept was then softened to a "day off", and on 24 October of that year an estimated 90% of the country's women downed tools, in both the workplace and the home. In Reykjavik, 25,000 women gathered for speeches, talks and singing – at a time when the entire Icelandic population numbered less than 220,000.

Thorunn Sveinbjarnardóttir, 45, was the country's minister for the environment between 2007 and 2009, and is now studying for a master's degree. She was 10 at the time of the original Women's Day Off, and went with her mother. "I just remember the feeling of being among this mass of women, who were all so happy," she says, as we sit in a cafe on Laugavegur. "That was a lesson for my generation, and I think the secret ingredient was that we managed to get women from all corners of society – from both the left and right, politically, and from all social classes. That was very important. It was a euphoric day."

The ability to mobilise women of all stripes – a really unusual feat – is still much in evidence. Last year Women Strike Back revived the spirit of the Women's Day Off, and despite storm warnings, 50,000 women flooded the Reykjavik streets, a third of the country's female population. (In the UK, it is considered a strong, successful feminist protest when 2,000 of the country's 30 million women come out.) Video of the event shows women in padded jackets, pink catsuits or Lopapeysa (traditional Icelandic jumpers), their hair, scarves and capes being whipped by the wind. One woman is dressed as a Viking. Some are laughing, many brandish signs, all look determined.

Out in the Reykjavik suburbs, I spend an afternoon with Sigrídur Magnúsdóttir, Andrea Halldorsdottir and Eva Gunnbjornsdottir. When I ask which women's issues upset them most Eva, 31, a postgraduate student, plumps for the pay gap, while Sigrídur and Andrea talk passionately about the problem of sexual violence. "If I could change one thing," says Sigrídur, 35, an office cleaner, "it would be the sexual crimes against children and women. Men will have to fight for themselves." Andrea, 27, a music teacher, says even in cases where someone is raped and almost left for dead, the reported punishment seems shockingly low.

I talk to Gudrun Jónsdóttir, a veteran feminist campaigner in Iceland, who works for Stígamót, a counselling organisation for victims of sexual violence. She says the country is certainly "a paradise of gender equality on paper", but that the reality doesn't quite match. Each year, Stígamót and the rape crisis unit at Reykjavik hospital work with around 250 women "but we can count the annual rape sentences on one woman's fingers".

She says there is still a huge problem with people's attitudes, "within the justice system, among the public, and with the women who come to our place, who are filled with shame and guilt". Last year the head of the city's sex crime division, Björgvin Björgvinsson, resigned from that position after a newspaper interview in which he said many rape victims had been drinking or taking drugs, and therefore bore some responsibility for being assaulted. In November 2010, he was reinstated.

So Iceland isn't perfect, but there seems to be the public pressure and political will to tackle its problems. The prime minister tells me the country has "a very strong and vocal women's movement, which keeps gender equality at the forefront of the debate. The movement has held the political system accountable to a degree where we can say that no politician who wants to be taken seriously can ignore the issue."

In its two and a half years in power, the government – a coalition of social democrats and left-greens – has been impressively active. It has criminalised the purchase of sex, introduced an action plan on the trafficking of women, and banned all strip clubs. When it comes to domestic violence, Katrin tells me, they have moved towards "the Austrian way", in which whoever committed the violence has to leave the home, rather than the victim going to a refuge. They have also introduced a law to take force in 2013, obliging corporations to have at least 40% of each gender on their boards.

Iceland has a history of progressive female politicians. Vigdís Finnbogadóttir, the country's president from 1980 to 1996, was the world's first democratically elected female head of state. At the time of her initial victory, the number of female politicians in the country was very low – just 5% of MPs – and so in 1983 the Women's Alliance was formed, an explicitly feminist party, which at its highest point, in 1987, held six seats, out of a total of 63. They fought for better wages for women, and, says Thorunn, who was a member, "spent the 1980s talking about all the taboos – rape, incest, domestic violence, putting in place legislation to protect women and children. All those issues are mainstream now, but it took a lot of courage."

In 1994, Ingibjörg Sólrún Gísladóttir, who had been a politician with the Women's Alliance for more than a decade, became mayor of Reykjavik, a position she held until 2003. And in 2009, after the financial crisis, and at a time when the country was questioning the values that had led them there – risk-taking and bravado, for example, which many defined as specifically masculine – there was much talk of women cleaning up the mess. Women were appointed to lead two of the disgraced banks, New Landsbanki and New Glitnir, and Jóhanna Sigurdardóttir became Iceland's first female prime minister. I ask Gudrun Jónsdóttir whether she thinks Jóhanna is a feminist, and she says: "Perhaps not primarily – she comes from the labour movement, she was a flight stewardess – but she's been around in politics for decades and has a great personal respect for the movement. Through the years she has supported our work as well as she can, and we really feel she is in our corner."

This support for feminism is not exclusive to the country's female politicians. This fortnight's copy of the Reykjavik Grapevine, an English-language paper, features an interview with the city's current mayor, Jón Gnarr. He and his party of artists and comedians – pithily called the Best party – describe themselves as "anarcho-surrealists" and were voted in last year, apparently on a tide of public animosity towards the country's establishment. But when he talks about women, as he does at length, he sounds serious. "I believe a healthy society must build equally on the forces of men and women," he says in the interview. "Our culture is just insanely male dominated, and we as a party wish to confront and change that."

Jóhanna is the world's first openly gay prime minister, while Vigdís, who seems universally beloved, was famously a single mother. Single motherhood isn't unusual in politics here; Katrin had her first child at 23, and raised him alone for 11 years, while building an impressive career. Parents here talk strongly of community support, of collective care for children, and there is no sense that motherhood precludes work or study, which effectively changes the whole structure of women's lives. "You are not forced to organise your life in the 'college-work-maybe children later' way," says Thorunn, who is a single mother to a young daughter. Andrea says when she had her first child, on her own, at 19, she took him with her to school, "and the teacher would hold him while I was studying".

Joanna Dominiczak, a teacher and chair of the Women of Multicultural Ethnicity Network, says that "having a child here is seen as a gift. You don't have to think, Oh my God, am I going to be able to afford one, two, or three?" The country has progressive rights regarding parental leave after a child is born, with "the mother having three months, which is untransferable," says Joanna, "the father having the same, and then the parents having three months they can share." This sets up the importance of both parents from the start, and skewers the discrimination endemic in many societies, including the UK, where women of child-bearing age are less likely to get jobs for fear they might at some point need maternity leave. (If companies chose to discriminate against both men and women of child-bearing age it would rule out most of the workforce.)

Sigrídur, Eva and Andrea are all single mothers, and while they have some grumbles, all are positive about the nurseries and schools their children attend. Annadís Rudolfsdóttir, studies director of the gender equality studies and training programme hosted by the University of Iceland, lived in the UK until recently, and says it's much easier to be a mother in Iceland. "It costs a fortune to put your children in a nursery in the UK," she says, "but here, as a single mother in Reykjavik, with your child in a nursery eight hours a day, you pay about £70 a month … If you're part of a couple, married or co-habiting, it's about £118 a month. You can imagine how much easier it is when you've got those facilities behind you." That includes breakfast and lunch. (It was recently reported that 32,000 women left their jobs in the UK last year, in large part due to the rising cost of childcare.)

There is a distinctive confidence to Icelandic women; it's been suggested their independence was forged through centuries of watching their male partners head off to sea, while they dealt with everything on land. Whether there's any truth in this, they certainly seem to have an innate toughness. When I ask a young woman in a sandwich shop if she believes Iceland is a feminist country, she says her colleagues have been discussing this, and their conclusion is: "You don't fuck with Icelandic women." She narrows her eyes behind her pink vintage glasses.

When I ask about the best aspect of being a woman here, the word that keeps arising is freedom. "You can do whatever you want," says Andrea, "you can educate yourself to be a pilot, a cop, anything you like." (In a cosmetics shop, I bump into the country's first female coastguard.) "We have a prime minister who is a woman, and our president used to be a woman, so we've grown up feeling anything is possible."

Katrin echoes this. "I've always had this freedom, as a woman in Iceland, making choices for me and my family, on my own terms. You're supported if you want more education, if you want to work. You could say we are quite a liberal nation." She looks out of the window, where the rain is still battering the building. "I love this weather," she sighs happily.