Swinging Strike + Hit Batter + Dropped 3rd Strike

[bfoster]

I've been looking through rules and online for a specific clarification on this situation as it occurred while umpiring a game yesterday.

Less than 2 outs, first base open, 0-2 count.

Batter swings at an inside pitch and "fouls" a ball off of his hand.

I called the batter out. His coach, first, thought the hand is part of the bat. I said "No". Then he said that if it was a strike, it wasn't caught, so it should have been a dropped third strike and the batter should have an opportunity to make it to first. This was a little less clear for me. But, i was pretty certain that the ball is dead, so there's no play to be made.

I was correct that the ball is dead the instant it hit the batter. But, I still haven't found any specific exclusion for a third strike not needing to be caught on a hit batter, dead-ball or not.

Common sense tells me that the batter is out. But, I would like to clear this up with a specific rule or set of rules. Any help?