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Wrong - The batter is out, and if you don't get back to 3rd quickly you could be doubled off for a 2nd out. To advance on a caught fly ball you must leave the base only after the ball is caught.
From the official rules:
Any runner shall be called out, on appeal, when --
(a) After a fly ball is caught, he fails to retouch his original base before he or his original base is tagged;