Ball four - runner stealing with throw to 2nd

[BEAREF]

Runner on first...3-1 count on the batter...on the pitch the runner is stealing 2nd...pitch is a ball...catcher throws to 2nd attempting to throw out the runner...BU calls runner out (not knowing that the pitch was called a ball)...runner assuming that she is out heads to third base dugout...3rd base coach tells her to stay at third because it was ball four and she's not out. After everything stops the umpires get together to discuss the situation. They place runners at 1st and 3rd.

Seems right to me, but would like a rule or casebook example to back their decision.

Thanks.