I have 1994 e36 is with the same problem. The wipers will not turn off. The problem would stop if I turned off the key and restarted. The next day the problem is full time. The wipers come on when you turn the key and will not go off. The windshield washer works, the different speeds work, but they will not turn off.
I unplugged the fuse to stop it but now the radio does not work.