Suppose you have a one parameter family of algebraic varieties over unit disk, such that the central fiber is singular and is a union (normal-crossing) of two varieties and the rest are smooth.

Is it true that every line bundle over a smooth fiber, after some base-change, extends to the whole family? Do you know any counter example?

This is the true (although not very obvious) for a family of curves, since line bundles are just bunch of points.

Extra assumption: $h^{2,0}=0$ for the smooth fibers, and they are simply connected, so that the ample cone does not move within $H^2$.