No, if you touch first base with the bat in your hands you are out.
The above answer is wrong. There is no rule covering this. A runner can carry the bat all the way around the bases without violating a rule.
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In Major League Baseball, there is no rule prohibiting the batter from carrying the bat to first base, as long as such action does not interfere with defensive play. I've never seen it done, but there is no rule specifically prohibiting it.
Other leagues may have rules addressing this question. You would need to ask a league official from your specific league.
If you make contact with the ball and your foot is complete out of the box you are out.