Suppose that $f\colon X\to Y$ is a proper (or even projective) morphism of (reduced) algebraic varieties over an algebraically closed field $k$. If fibers of $f$ over all closed points of $Y$ are reducible, does it imply that the generic geometric fiber of $f$ (that is, the pullback via $\mathrm{Spec}\,\overline{k(Y)}\to Y$) is also reducible?

Thank you in advance,
Serge