I've shot a 20 gauge shotgun and a 12 gauge shotgun. The 20 gauge didn't have much kick at all, and I could imagine an 11 year old girl being able to shoot one. It's generally what's recommended for younger kids in skeet tournaments.

However, the 12 gauge had a pretty wicked kick, and I know how to hold and fire a gun. Perhaps it's possible for an 11 year old girl to shoot one, but I'd really have to see it to believe it.