If $\mathbf{D}$ is the complex unit disc with coordinate function $s$ and $X \to \mathbf{D}$ is a proper flat holomorphic family (and it is smooth outside of the fiber $s=0$), will the total family $X$ deformation retract onto the fiber above $s=0$?

I don't think this will be true, but I can't find a counterexample. I thought that if $X$ is a family of elliptic curves degenerating to a singular cubic, then this would fail. But, I think it actually works in this case.

Edit: in the comments, this is claimed (without proof) to be true.