Assume $K$ is a complete discrete valuation ring, with valuation ring $A$ and maximal ideal $\mathfrak p$, $E$ a finite extension of K and $B$ is the integral closure if $A$ in $E$, $\mathfrak B$ the unique prime lying above $\mathfrak p$ in $B$.
Is it always the case that $B/\mathfrak B=A/\mathfrak p(b)$ for some $b$ in $B/\mathfrak B$? Or we need conditions like $A/\mathfrak p$ perfect? (Is $E$ separable over $K$ enough?)
No it is not always the case: take a finite inseparable extension $l|k$ possessing no primitive element and consider the ring extension $l[[t]]|k[[t]]$ (rings of power series in $t$). The extension $l((t))|k((t))$ of the fields of fractions is inseparable too and the corresponding extension of residue fields is $l|k$.