No, that is not possible. That thought comes from the fact that, in certain situations, a batter can strikeout but get on base if the catcher does not catch the ball on strike three and cannot get the ball to the first baseman before the batter reaches first base. This is called the 'Dropped Third Strike Rule' and there is a link to its description on this page. There have been many occasions, I don't know the exact number, where a pitcher has struckout four batters in an inning. This can only occur when the catcher does not catch strike three on a strikeout and the batter reaches first base safely.
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