Swinging Strike + Hit Batter + Dropped 3rd Strike

[Dave Reed]

From the 2009 Casebook:
5.1.1 SITUATION E: On the third strike, B3 swings at and misses a pitch. The ball touches his arm or person. RULING: B3 is out. The ball becomes dead immediately. (5-1-1a, 8-1-1d)

From the rules:
5-1 ART. 1 . . . Ball becomes dead immediately when:
a. a pitch touches a batter or his clothing (8-1-1d), a runner (8-3-1a);
1. The ball becomes dead even though the batter strikes at it (8-1-1d)......

and,

8-1 ART. 1 . . . A batter becomes a runner with the right to attempt to score by advancing to first, second, third and home bases in the listed order when: .....
d. a pitched ball hits his person or clothing, provided he does not strike at the ball; ....

It's the last phrase that matters--"provided he does not strike at the ball". So the batter who swings and is hit by a pitch doesn't become a runner. He does however, have a third strike charged against him, and he is out:

7-2 ART. 1 . . . A strike is charged to the batter when:....
b. a pitch is struck at and missed (even if the pitch touches the batter);