Yes, the batsman is out.
Refer first to Law 35 (Hit Wicket)
Out Hit wicket
(a) The striker is out Hit wicket if, after the bowler has entered his delivery stride and while the ball is in play, his wicket is put down either by the striker’s bat or by his person as described in Law 28.1(a)(ii) and (iii) (Wicket put down).
We need to check Law 28 to check the exact details of what constitutes 'Wicket put down'
Wicket put down
(a) The wicket is put down if a bail is completely removed from the top of the stumps, or a stump is struck out of the ground,
(i) by the ball,
or (ii) by the striker’s bat if he is holding it or by any part of his bat that he is holding,
or (iii) notwithstanding the provisions of Law 6.8(a), by the striker’s bat in falling if he has let go of it, or by any part of his bat becoming detached
That last line "or by any part of his bat becoming detached" covers the situation in the question. The batsmen is out.