Could you have whiplash from simply falling over?

Probably not: That would be very unlikely. You can injure the neck, but most would not consider it to be a whiplash injury.

It's possible!: Typically, whiplash happens when the forward motion of your body is suddenly stopped and your head keeps accelerating forward and then snaps backwards. It is possible to have this happen when you fall depend on how you fall. Either way you can have the same degree of neck pain.