Can STM tunneling happen faster than the speed of light?
No, a tunneling electron does not move faster than the speed of light. The classical picture of the electron having a localized trajectory, and taking some time to travel from one well to the other, must be abandoned. When the tunneling happens, the wavefunction of the electron is already extended partially into the adjacent potential well. Physically that means that the electron just has a probability to exist in the tip well rather than surface well.