Is it a fair ball, hit batter, or strike?

Question
Thank you for the answer.  In the situation described, the hands were still on the bat during the swing attempt.  Does that have any bearing on the ruling?  And, are the hands ever considered part of the bat, regardless of the count, and during a swing attempt or no swing attempt?

(Ironically, I saw a similar situation had occured to a Texas Rangers batter in a MLB game.  The pitched ball hit the batter's arm instead of the bat, during a swing attempt.  I didn't know the strike count, but the ball was ruled a fair ball.)
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Followup To

Question -
With 2 strikes, while making a full swing attempt at an inside pitch, the batter is hit on the hand.  The ball rolls into fair territory similar to a bunted ball.  The batter still in the batters box, drops the bat and grabs her hand in pain.  The umpire immediately ruled a dead ball - strike 3.  The umpire made these comments when I questioned the call.  The pitch was not in the strike zone. The hands are not considered part of the bat during a swing. The swing occured first and took precedence over the ball hitting the batter - the batter is out and is recorded as a strike out.  Is he correct?  Should it be a fair ball, dead ball - hit batter, or strike 3?

Answer -
Hi Clay,

By rule the batter is out when the third strike is swung at and the pitched ball touches any part of the batter's person.

The umpire had the correct call.

Mark

Answer
Hi Clay,

The hands are not considered part of the bat.

If there was no swing and the ball is in the strike zone and hits the batter's hand it is a strike.

If there is a swing and it hits the batter's hand it is a strike.

Both are dead balls

Let me know if you need anything more

Mark