Never Seen This Happen Before — What's The Ruling?

[The Stiff]

I have never seen this happen in almost 40 years of coaching. The umpire ruled promply and definitively, but I have no idea whether the ruling was correct or not. Here is the play:

One out, runner on third. Batter swings and misses and strikes out, with the ball striking her arm and going to the backstop (catcher misses). Runner scores, batter reaches first. Umpire rules that the run can score, but rules batter out.

Seems to me that 1) ball should either be dead, no one can advance; or 2) ball remains alive and everyone can advance. Don't understand why one runner can advance, but not the other.