If you'd ask me this question about 11 years ago, I would have answered 'yes'.
I had a '64 Austin Mini 850 Deluxe in red, mostly original paint, original interior & engine, running.
I guess not many were interested in these things back then...it eventually sold for RS. 19k (yes, 19,000) to an Afghan couple who intended to drive it back to Kabul.
Finding one, particular a Cooper, in unmolested condition is pure pot luck.