What is the function of the ASP BUTTON in a Mini Cooper?

I'm sure that is described in the owners manual. I think you mean ASC button. That is the acronym for Automatic Stability Control. Turning it off will disable it. The ASC system uses speed sensors at each wheel and throttle position to improve traction by applying the rear brakes if the rear wheels spin faster than the front. It can also shut off individual fuel injectors and close the throttle valve to help maximize traction provides traction control by modulating throttle, ignition timing and braking force to maintain traction. I would advise to not turn it off unless you were stuck or wanted to have some spirited driving.