On a third strike if the ball gets away from the catcher the batter can try to run to first. If he beats the throw to first he is safe and the inning continues even though the batter is officially “out”.
Well, the official scorekeeper marks that as the batter bing on 1st base via K-e2 (Strikeout, batter reaching first on an error by the catcher).
The batter is credited on his stats with a strikeout (not an Out).
Not true that the hitter is officially “out”. He is not an “out” even though the pitcher is credited with a strikeout. So there can be more than three “strikeouts” in an inning, but never more than three “outs” in an inning.