Although they're rare, I've seen a few instances where a baseball player breaks their bat and still manages to hit a home run. Two examples:
- Chris Davis from the Orioles did it in 2012 (https://www.youtube.com/watch?v=mJ18aGmKnc4)
- Bryce Harper from the Nationals did it in 2018 (https://bleacherreport.com/articles/2770901-bryce-harper-launches-406-foot-broken-bat-home-run-during-nationals-vs-mets)
I feel sure there are other instances, but this will give the idea.
Broken bats are not especially rare, but usually any hit resulting from a broken bat is weak. When the bat shatters or the head breaks completely from the handle it is not unusual that the fragments of the bat may go farther than the ball. When the bat cracks but does not fully break, it's not unusual for the ball to make it to the outfield, but these hit usually do not carry as far as if the bat stays intact.
In the two examples above, however, the bats snap completely, leaving the player with about 8-12 inches of handle in their hand and the ball carries over the wall, so let's say greater than 375 feet. The Harper home run was estimated at 406 feet.
The physics of these broken-bat home runs is not obvious to me. Once broken, the player cannot provide torque on the head of the bat, so they should have much less ability to change the momentum of the ball, and some energy must go into breaking the bat. I saw the Chris Davis home run above when it happened on television, and in the slow-motion replay, as I recall it, it also wasn't clear when in the swing the bat broke. I got the impression that it might have broken after contact with the ball was complete. (The announcer says the bat head ended up "in the Oriole's dugout," which is a clue to where it became disconnected from the bat. The dugout is more or less behind the left-handed hitting Davis in foul territory, and the ball cleared the wall in fair territory. Likewise, the announcer on the Harper home run says the bat head "hit the screen," which suggests that it flew reasonably far into foul territory while the ball cleared the wall fair.)
Is there some vibrational mode or other failure mode in the bat that would allow it to break in the follow-through? Could it be that the bat head already has sufficient momentum to redirect the ball, so the loss of connection to the hand doesn't matter? I think the former is more likely, but I'm not sure how that would manifest in the bat.