No. (NFHS citations below)
9-1-1(a), Exception -- A run is not scored if the runner advances to home plate during action in which the third out is made as follows:
a) by the batter-runner before he touches first base.
8-1-1(b) -- 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:
b) he is charged with a third strike;
1. If the third strike is caught, he is out an instant after he becomes a runner.
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