Is there a compact, connected, smooth 3-manifold $M$ with boundary $S^2$ not diffeomorphic to $D^3$ (the closed unit ball)? If so, what is it?

The compactness condition rules out the complement of the open ball in $\mathbb{R}^3$, and the connectedness condition rules out the closed unit ball disjoint union with something having empty boundary.

~~My only idea is to consider things that have boundary $\mathbb{RP}^2 \cong S^2$, but this hasn't gotten me anywhere yet.~~

If you have an example, some explanation of how you thought of it would be great.