MIA is the most provocative pop star of her age. Here, the rapper discusses her new multifaceted album, the absence of her father during childhood and having the video to her latest song, in which a child is 'murdered', censored by YouTube

MIA might not be properly famous – not yet, anyway – but she is notorious. Her songs are both played on the radio and banned by MTV. Her political statements have led to her being described as a terrorist. Her pop videos are analysed for propaganda; the last one she put out, in April, was immediately removed from YouTube for its violence. And in an age where "immigrant" is an insult rather than a description, she uses the word proudly to describe herself: a refugee who started from nothing and became internationally successful by the time she was 28.

She's also a drama queen. "This," she announces at the end of our chat, "may be the last interview I ever do." Don't put money on it. MIA – real name Maya Arulpragasam – is a gift that the media will not let slip away. Beautiful, with flashing eyes and jutting chin, her wild, scattershot style – she favours bright colours, African and Asian prints, hoodies, all-in-ones and has modelled for Marc Jacobs – is accessorised by wild, scattershot statements. She takes up causes ("I think I'll talk about Liberia and stuff") without much thought and picks fights apparently at random: recent targets include Lady Gaga and the Twilight films as well as Facebook and Google, which, according to Maya, were "developed by the CIA".

Mostly, she gets into trouble for speaking out about Sri Lanka. Born in London in 1975, she moved to her parents' native Sri Lanka when she was six months old, spending her first 10 years in a village outside Jaffna, in the north, and in Chennai in India. Her father was a founder of Eros, a student body which campaigned during the 70s and 80s for a separate Tamil state; his actions meant that Maya and her family were forced to return to London in 1987, where they lived in bedsits, hostels and council flats.

Maya doesn't take her politics from her dad, but she does speak out in support of Tamil citizens. Her assertions that the Sri Lankan government is guilty of the genocide of local Tamils have not been going down well. She's even been accused of supporting the LTTE, or Tamil Tigers, the separatist group that was defeated by the Sri Lankan government last year, ending almost 30 years of civil war. Kylie she ain't.

Aside from all that, in the month before we meet, MIA causes two big fusses. The first is over "Born Free", a track from Maya (or /\/\ /\Y/\ as it is written), her forthcoming third album. Without consulting her record company – "Born Free" is not a single – Maya and director Romain Gavras decided to make an accompanying video, funding it themselves.

Gavras comes with his own notoriety: his video for Justice's 2008 track "Stress" shows a teenage gang on the rampage in Paris, smashing up cafes, beating up security guards, robbing a car and torching it. And his nine-minute-long clip for "Born Free" proved to be even more provocative. It starts in LA, with armed militia rounding up several red-haired young men and one child; the detainees are taken to the desert, where they are beaten up and killed. The boy is shot in the head. The metaphor is obvious, but the illustration is graphic, some would say gratuitous, and the video was immediately banned by YouTube, though you can still see it on miauk.com.

The second furore is over a piece in the New York Times by Lynn Hirschberg, where the writer emphasised the contrast between Maya's political pronouncements ("My giving birth was nothing when I think about all the people in Sri Lanka that have to give birth in a concentration camp") and her now-comfortable lifestyle. To be honest, for the waspish Hirschberg, the feature wasn't that nasty, though its tone sneered and she made many of her points through implication. For instance: '"I kind of want to be an outsider," she [MIA] said, eating a truffle-flavoured French fry."'

Maya was not pleased with such an insinuation. So she posted Hirschberg's phone number on the net, along with a recording that proved it was Hirschberg, not Maya, who ordered the posh chips. A storm on a dinner plate? Maybe. But Maya – unsophisticated, sincere, feisty Maya – just can't let things slide.

I am stuffed full of MIA knowledge by the time we meet: my computer fizzing with arguments for and against, statements from governmental spokesmen, rants from barely literate commentators, Maya's own, capital-lettered tweeting and blogging. It's all very entertaining – but it does distract from her music.

So, let's be clear: MIA's music is fantastic. A self-taught rapper, songwriter and producer, she makes infectious, inventive tunes that smash and grab from hip-hop, grime, rave and punk, stir things up with Jamaican beats, Indian rhythms, African drums. She samples with abandon, from older artists such as the Clash and Suicide, as well as from the sounds that surround her: cash registers, toy instruments, the Tamils' urumee drum. Her lyrics are interesting, too: fluid, ambiguous, hard to decipher, they mix up gangsta talk – "Some some some I murder" – with patois, London slang, silliness, personal observation. After just one LP, 2005's Arular, rapper Nas commented: "Her sound is the future."

We meet in the Banana Leaf, an Indian restaurant at the grottier end of Tooting in south London: Maya chose it because it's round the corner from her mum, Kala. When I arrive, the restaurant is quite empty, apart from Maya, her press officer, her boyfriend, Ben Bronfman, and their 14-month-old son, Ikhyd, who's happily trotting around, searching out mobile phones to chew. Everyone else is picking over the remnants of a big meal. Having girded my loins for rent-a-gob hour, I find quite the opposite. Maya, Ben and Ikhyd seem almost sleepy: they've been doing a lot of travelling.

Ben, a relaxed, hood-eyed 27-year-old, offers to take Ikhyd out for a walk while we talk. Somehow, before he does, we get on to the topic of scuba-diving, which Ben and Maya recently tried. Ben loved it, Maya was a bit scared. "I can't do stuff like that!" she says. "Posh people can though," she continues, pointing her fork at Ben and laughing.

Part of Lynn Hirschberg's sniffiness in her New York Times piece was based around Maya hooking up with Ben, whose dad is Edgar Bronfman Jr, the Seagram's heir and CEO of Warners. Ben, the lead singer of indie-reggae group the Exit, runs his own, environmentally friendly record label and was born to wealth. Maya, clearly, was not. The implication: how can she still keep it real when she lives in West Hollywood with a trust fund kid? But such class-bound objections seem ludicrous when you see Ben and Maya together: their backgrounds are different, but they are both music types – good-looking lefties in nice trainers and cool clothes – and operate in that turn-and-turn-about way of any family with young kids. Ben is clearly very proud of his wife-to-be. "Have you heard the record?" he asks keenly. "Isn't it great?"

Perhaps it's because Ben and Ikhyd are around but, after they leave, Maya and I talk for a long time about her family. (She speaks in a strange, drawn-out, London/US drawl.) It's a wild ride of a tale. Though Maya's mum and dad were both Tamils from Sri Lanka, they met for the first time in a pub in Hounslow. Arul, her father, had landed a scholarship to learn engineering in Russia when he was 15, after which he came to London; Kala was studying for a few months, staying with her brother. Kala needed to extend her visa, Arul agreed to marry her, and did so, in a matter of days. They had two girls in two years, Kali (now a jewellery designer) and Maya, short for Mathangi. But unknown to Kala, Arul had become involved with some politically minded Tamils, and, when Maya was two months old, he left. "He went out to buy a pint of milk and didn't come back for four months," says Maya. He went to Lebanon. To train with the PLO.

When Arul did return, it was to inform Kala that he was moving back to Sri Lanka and did she want to come? Influenced by his new friends and his time in Russia, where, according to Maya, there was a kind of coffee-table radicalism ("Every cafe had typewriters and you would think of an ideology, type it up, photocopy it, hand it out to a thousand people. Within two hours you had a rally"), Arul Pragasam was off to his home country to help set up Eros (the Eelam Revolutionary Organisation of Students). Eros was devoted to supporting the Tamil minority through politics but not violence. It was disbanded in the mid-80s when the Tamil Tigers became the stronger force.

Anyhow, Kala moved back with Arul, to her home village, near Jaffna, in the north of Sri Lanka, where Maya's grandparents could help out. Soon after they got there, son Sugu was born. However, Arul, who now called himself Arular, was still not around. "He asked my mum, 'Why would I devote myself to one woman and three children when I could be helping thousands?' She said: 'If you even have to ask that, you should go.'" Until she was eight, Maya believed her father was dead. She saw him three times in the years she was in Sri Lanka. "One of those times, when he came home, he didn't even know what I was called," she says.

Maya's life in Sri Lanka was happy: she, her sister, brother, mum and grandmother lived in a three-room unit (granny slept in one room, everyone else in another and the third was for visitors). On either side of their house were their aunties, with seven and eight kids apiece, "and on our street there were maybe 50 kids: it was brilliant". Maya's family was the only one on the block with a fridge – "known as the cridge" – and, once a month, they would hire a TV and video. Everyone they knew would come over, cram into one room and watch Tamil films for 24 hours nonstop. They'd put ice on their faces to keep awake.

But Arul still managed to affect their lives. Her family moved again, to a Tamil area of India, for three years, where they lived in a derelict house. When they returned to their Sri Lankan village, government soldiers kept turning up to find him; they would sit Maya on their knee, ask her if her toy was a present from her dad – "I would be, like, I wish!" The same friendly soldiers would then beat up Kala in front of her children, and mete out the same punishment to any local male between 18 and 25. "Kids like my cousin, and all he did was hang about, whistle at girls and sing Michael Jackson songs." After some years, this harassment became too much and the local community scrabbled some money together and gave it to Kala, telling her that she needed to take the kids and go back to the UK.

For an absent father, Arul caused a lot of grief, I say.

"Even now, really," says Maya. "Because everyone thinks my story is to do with my dad, when, you know, it's my uncle in Morden [south London] on my mother's side who's my inspiration." And she launches into his tale: he smuggled himself into the UK, sold clothes out of a car, ended up in the 1960s as "the first ever brown guy to have his own stall on Petticoat Lane". Everywhere you look in Maya's vast family, there's a story of adversity overcome, an epic adventure.

After she returned to the UK, aged 10, Maya heard nothing from her dad, until she summoned him back into her life by calling her first album Arular. "I thought that if he Googled himself, he'd get my LP and then he'd get in touch." The tactic worked, but their relationship is still fraught. Maya learnt most of what she knows about her dad from other people. "It irritates me that I end up giving him so much attention when he had so little to do with my life," she says.

All the while that Maya is talking, I think: she's nothing like I expected. She's gentler, more smiley, more discursive. Her stories ramble sideways; she explains rather than rants. Still, it would be easy to pull out a damning quote or two; she's no diplomat. Talking about Brick Lane, where she hung out when she was 17, she says: "Tower Hamlets got a new MP and because he wanted to tone down the violence in Hoxton, he flushed the area with loads of heroin to sedate and pacify the Bengalis."

It's hard to believe that an MP would do that, I say, mildly.

"I think when a housing community is going to explode then, ultimately, someone gains, whether politically or financially. All the buildings I was hanging out in, the leather factories, the community centres, they all got sold and turned into expensive condos. It's all connected."

Such conspiracy theory thinking is common enough. In fact, Maya's politics, other than her Sri Lankan ones, which are more informed, remind me of Banksy's: essentially, people with power are all corrupt and they conspire to keep the innocents down. She may well believe these things, but you need detail to stand up such claims and Maya will never win a Private Eye award for investigative journalism. That's not what she's for. She's a provocateur.

Which brings us to the video for "Born Free". Maya thinks the reaction to it was ridiculous, given that YouTube has plenty of real-life killings on it. "It's just fake blood and ketchup and people are more offended by that than the execution videos," she sniffs. By execution videos, she means clips of Sri Lankan troops shooting unarmed, blindfolded, naked men in the head. The Sri Lankan government claimed the videos were fabricated; independent experts have confirmed them as genuine, probably made in January 2009. "I Twittered about them six months ago and no one talked about that, and then me and Romain make a video and everyone's like, 'Oh my God.'"

Maya's problem is not that she's uninformed, but that she's emotional and personal and, like, you know, um, a bit inarticulate. You're not meant to get involved when giving out information about war. After our interview, I watch a clip of her on US chatshow host Tavis Smiley's programme, where she very clearly differentiates between the Tamil Tigers and the Tamil citizens. She then says that the Sri Lankan government has been using the Tamil Tigers as an excuse to wipe out ordinary Tamil civilians. She gave the interview in January 2009 and later her version of events was confirmed: it was reported in the New York Times.

"The New York Times," says Maya, "is such a big institution that the journalists don't even read their own news. But two days before Lynn's story came out about me, there was a New York Times news story saying that the Sri Lankan government forces are to be blamed for all the civilian deaths. Yet the day I had my baby and was in hospital coming round, they'd printed a story saying that I was a Tamil Tiger sympathiser. From day one, what I've been saying is that I'm here for the people, talking about the citizens, not the Tigers. I don't know the Tigers, I don't know what they do, I don't give a shit. It's about the Tamil people, because I only know it from that experience.

"Also, when the New York Times put Sri Lanka's beaches as the number one destination [in a new year list of the 31 best travel ideas for 2010], six months after 300,000 people get bombed there... Tourism needs to be connected to politics. All wars are fought over land and you're advertising a piece of land as the best place to go and lay on and sunbathe. [You should] research that piece of land!"

All of which, to those of us uninformed about Sri Lankan politics, seems complicated. But I checked out Maya's accusations, and they all turned out to be true, though the piece that acknowledged that the Sri Lankan government shelled civilian safe zones came out nine days before Hirschberg's, rather than two.

The difficulty for MIA is not that she's lying. It's that the world doesn't really care. She spends a long time explaining to me how the Sri Lankan government dispersed negative propaganda about her on the internet, contacting her fans individually, because they wanted to present a clean image of their country. This may well be true, but if I'm honest, before researching for this piece, I wasn't really aware of the details of the civil war in Sri Lanka, not least that it had ended with hundreds of thousands of Tamil civilians being herded on to beaches and bombed. I'm just as guilty as Hirschberg. No wonder there's more shock about a silly pop video than real people really dying.

Maya has spent much of her life moving around the world, latching on to different groups of people. Bengali boys from Brick Lane, art students at St Martins (she talked her way in), LA hip-hop kids (she moved there when she was 18 for four months), or London's in-crowd (she shared a flat with Elastica frontwoman Justine Frischmann and fashion designer Luella Bartley in her early 20s). When she made Kala, her second album, she had problems with her US visa, so she recorded in Liberia, Jamaica, Trinidad, India. When she was pregnant, she had to move out of New York, because the smell of it made her throw up all the time. She doesn't quite fit anywhere.

She and Ben are getting married soon – they have 10 months before Maya's American work visa runs out again – but they don't know where, as their families are so dispersed. After that, they're thinking of living on a boat. "I'm so up for that," says Ben, who's back with Ikhyd.

Maya tells a story about moving back to England when she was 10. On her first day at school, her class were working through a sum. Maya put her hand up, because she knew the answer. "And literally the whole class turned round and laughed at me," she says, laughing herself.

The teacher patted her on the head and told her she didn't have to pretend. But she did know the answer – she just couldn't speak English. She didn't have the words to tell them.

"I didn't care," she says. "I was happy. I was happy wherever."

Maya is released on 12 July on XL