No extra alloy, no full size spare tire, sometimes no spare tire at all.
In the US, the non-S Cooper will come with a space-saver spare tire as a standard feature - but only if fitted with 15 inch wheels. Upgrading the Cooper to 16 or 17 inch wheels means run-flat tires which means no space-saver will be provided. Perhaps one can be purchased and fitted by the owner? Or would its use confuse the ABS and other systems?
For the Cooper S, there will be no spare fitted - only run-flats. No chance of buying a space-saver for the S as the battery sits under the rear hatch floor.
Oh, almost forgot, in some countries MINI are provided with a thing called the MINI Mobility System which I believe is a package that contains an air compressor and a bottle of tire goo to temporarily fix a flat tire.