Dropped third strike question

[okmitzi]

this question applies to little league, and i know that rules vary, but i would just like a "generalized" answer to my question. the batter takes a called third strike, catcher drops the ball. batter hesitates for a second, someone yells "run!" and off he goes, reaching first base. the batter made no step back towards the dugout, in fact he didn't even leave the batter's box. the umpire calls him OUT saying he "hesitated" before running to first. what i have found so far indicates that the batter actually has until the next pitch in the game to run to first unless the umpire has called him OUT. does anyone know the rule? thanks in advance.