In a game last night, a thrown ball, hit defensive coach who is allowed to stay in the field (league rule).After the game my partner said it was a dead ball situation because rule 3.15 does not cover defensive coaches on the field. I replay: if he is allowed to stay in the field ball must be ruled live. Am I right?. Is there something on this regard in the umpire manual?. Please advise

You won't find anything in a manual on this because it resulted from a local league rule. No rule covers a defensive coach being on the field during play because it is not anticipated. If a thrown ball hits an offensive coach it would be just a live ball if he did not intentionally get hit and was not in fair territory. If he was, it would be dead and you'd get an out.

Common sense would say you should leave the ball live in your situation, however. After all, a defensive player threw the ball and hit his own coach. Why should the offense be penalized? I can see a runner about to score when F7 rifles the ball into his coach standing behind the shortstop position. Time...runner is returned to 3rd. Come to think of it, it might be kind of fun. They use to be allowed to throw at the umpires to kill the ball, so why not!

Actually, if you're in chartered Little League, Reg XIV(d) "Exception: in Minor League and Tee Ball, managers and coaches may be on the field for instructional purposes, but shall not assist runners or touch a live ball. At least one adult manager or coach must be in the dugout at all times."

As they are thus authorized - apply 3.15.

how I would call it
 

Assuming that the ball hitting the coach was accidental, I would keep the ball live with no penalty assessed. If it was intentional, the ball would be ruled dead, and the penalties for defensive interference should be enforced