Let $f\colon A\longrightarrow B$ be a finite extension of local rings and suppose $f$ flat. We know that under these assumptions (Thm. 2.16 Algebraic Geometry and Arithmetic Curves, Qing Liu ) $B$ is a free $A$-module. Let $\mathfrak p\subseteq B$ be a prime ideal. Denote by $\mathfrak p'$ the contraction $f^{-1}(\mathfrak p)$. I am wondering if $B/\mathfrak p$ is free as $A/\mathfrak p'$- module. Any help is well accepted.
Remark: In general, the ring map $A/\mathfrak p'\longrightarrow B/\mathfrak p$ induced by $f$ is not flat.