If you unplug the switch and it returns to normal, your problem is with the switch. Either in adjustment or it's faulty.
Between the switch and the brake pedal, there should be a spring. Unhook the spring from the pedal side and see if the light goes out. If it stays on, it's a bad switch. If it goes out, reconnect the spring and adjust the switch by turning the little thumbnut on the switch towards the switch itself. That will basically move the switch closer to the pedal, making the switch disengage.