Errrrrrrm yes....
I own and run a 94 1.3 SPi Mayfair.
Mini's came with the factory unoptional extra of rust and oil leaks. The engines are pretty much bullet proof and will even run on 1 cylinder...
Things to watch for are - rust - rust - rust and rust..
The rear subframes tend to rot, whereas the fronts are ALWAYS covered in oil, so tend not to..
The petrol tanks are very small (£20 to fill from empty) and will last around 180 - 200 miles but all depends how you drive it... I drive mine like I stole it TBH...
All depends what you want to pay... You wont get a road legal/road ready Mini for less than £1500 these days, if you do - be prepared to work on it...
Watch for the chancers who think that they have a Cooper and will even try to tell you that its a Cooper...
Also watch for it being re-shelled too. Not hard to do but an original Cooper (Mk1, Mk2 or Mk3) will command a high price and will be expected to be original. If its been reshelled into a more modern shell, they will still try for the higher price but it aint worth anywhere near what they ask....
Any questions, drop me a PM