In the case of the abysmal 2003 version yes, but in the case of the classic & superior 1969 original no.
No, because the original Minis original owner (BMC or British Leyland as it became called) weren't interested and didn't want their product associated with criminals and used in a gold heist.
So after much negotiating and persuading, Paramount films were only given 3 'Mini Cooper S' cars at cost price, the other dozen or more they required were paid for in full. Plus they bought a further dozen or more second hand basic Minis from a foreign car dealer simply to throw them off the cliff at the end. They wanted around a dozen or more, so they'd get all the shots they required.
In comparison, Fiat offered them all the cars they wanted for free, a Ferrari and $25,000 if they'd switch from using Mini Coopers to Fiats instead.
But the film was written with the Mini Cooper S very much in mind and it just wouldn't have been the same.
Oh and that's why if you look at the end credits, only 'Fiat' get a thank you for their support. BMC/British Leyland (who produced the Mini) don't get a mention.
All the facts appear in the book 'The Making Of The Italian Job' written by Matthew Field and sadly once the 1960's ended, so did the cool and optimistic feel Britain had.
After that it was all power cuts and industrial action and we've never had much to shout about since. I sometimes wonder why I'm still living in this miserable country, because it's pretty horrible now and yet we had it all back then.