Probably not. The elevator has security system just like any device. It will automatically trigger breaks in case it detects some sort of free fall.
Person inside free falling elevator could experience 0G or no gravity effect something similar to what astronauts experience in earth's orbit.
The only damage that could happen to you is when the breaks are released you won't feel 0G anymore and fall about 4ft to the floor of the elevator.
Hope this helps.
r3t40
