Benicio Del Toro must feel like Cannes is his playground.

It was here that saw his career truly take off when Bryan Singer's classic crime drama The Usual Suspects took the world's most prestigious film festival by storm in 1995. He returned for Fear and Loathing In Las Vegas and Sin City, before claiming Best Actor at the festival in 2008 for his stunning portrayal of the Cuban revolutionary in Steven Soderbergh's Che. No wonder when I ask if Cannes holds a special place in his heart, he mumbles "Hell yeah".

Dressed in a sharp black suit, navy polo shirt and tinted sunglasses, the Puerto Rican-born actor is back on the Croisette this week playing another real-life character, albeit slightly less known than Che Guevara. The film, Jimmy P (Psychotherapy of a Plains Indian), stems from a book-length case study, Reality and Dream, in which a psychoanalyst, Georges Devereux (Mathieu Amalric) took on a patient, James Picard (Del Toro), a Blackfoot Indian suffering from debilitating spells of temporary blindness and dizziness.

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Arriving in the wake of such high-brow therapy-related efforts as Paul Thomas Anderson's The Master and David Cronenberg's A Dangerous Method, it's a challenging work which some critics in Cannes balked at. Yet after industry paper Variety hailed Del Toro's "impeccable performance", it would be foolish to bet against him picking up a second Best Actor prize this Sunday (although he has stiff competition from Oscar Isaac, the lead in the Coen Brothers' folk-singer tale Inside Llewyn Davis).

It's not the first time Del Toro's played a Native American; he also did in Sean Penn's The Pledge, the film that convinced Jimmy P's director Arnaud Desplechin he was right for the role. Sensitive and assured in its handling of its subject, Del Toro felt the "importance" of telling a story about "original Americans", a phrase he's borrowed from his co-star Misty Upham, who plays Jimmy's wife Jane and is of Blackfoot ancestry herself. "There should be more stories about the original Americans," he adds.

In the US, Del Toro has already been criticised in some quarters for playing a Native American, some even equating it to John Wayne playing Genghis Khan in 1956 flop The Conqueror. Upham disagrees. "Benicio may not say it himself but he is making a very big step for Native American people," she tells me later. Del Toro simply shrugs, playing down any expectations he might have for the film. "It would sound very pretentious to say that movie said everything about Native Americans or is going to change the way they're treated in the United States."

The son of two lawyers, it's hardly the first time Del Toro has got political; Soderbergh's Traffic, for which Del Toro won an Oscar, tackled the war on drugs. And Che dealt with America's thorny relationship with Cuba, though the indifferent critical response – or "silence" as he puts it – in the US left him disappointed. Does he think there was a political agenda to those reviews? "I can't say 'There is the gun and there are the bullets', but there are some holes," he retorts, somewhat cryptically. "But it definitely wasn't the reception that it got here in Cannes."

Now 46, Del Toro has rarely been one for the Hollywood blockbuster (aside from his ill-advised hirsute horror The Wolfman). "I want to keep doing independent movies," he says, noting he will play a "cameo" in Paul Thomas Anderson's forthcoming Thomas Pynchon adaptation Inherent Vice. He's also just finished playing Colombian drug baron Pablo Escobar in Paradise Lost (as has, ironically enough, his aforementioned Cannes competitor Oscar Isaac, in a rival film). Is he expecting another controversial reaction? "I just expect that it's good. I hope that it's good."

Certainly, Del Toro is hoping to pass his integrity along to his young daughter, Delilah, who sprang from a brief liaison with the actress-model Kimberly Stewart, daughter to Rod Stewart. While she and Del Toro are not a couple, he was present at the birth and seems delighted that "my girl was born with her own personality – not her father's".

So is there anything he wants to teach her? "She's got to learn that you've got to work," he says. "It ain't gonna be about fame." In which case, he's already a model father.

Jimmy P (Psychotherapy of a Plains Indian) will be released later this year.