ASA Rule 8-1B says clearly that the batter becomes a batter-runner if the catcher fails to catch the third strike and there are 2 outs.
There is an exception for 10U, but otherwise, the batter may attempt to reach 1st base on a dropped 3rd strike with 2 outs.
With fewer than 2 outs, 1st base must be unoccupied.
But with 2 outs, the batter is not out and may attempt to advance.
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Tom
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