Is "flies off" natural when talking about a cut off arm

Yes, you can use "flies off" with things like an arm. Note that this does imply it's moving away with some speed, probably upwards (or at least parallel to the ground), and will probably end up some distance away. (From my memory, lightsabers don't generally result in that kind of motion when cutting somebody's arm, so that may not be what you meant.)

Alternately, if the arm simply fell to the ground (or landed nearby, with perhaps some horizontal motion, but still a motion that's more downward than upward/outward), then it would be more natural to say "falls off" instead.