Let $A$ be a Dedekind domain, $K:=\text{Frac}(A)$ and $L/K$ finite so that the integral closure $B$ of $A$ in $L$ is Dedekind. If $A$ is a PID, for example, then there exists an integral basis : $B$ is free over $A$, and this basis gives a basis of $L/K$. In particular, if $A=\mathbb{Z}$ or $\mathbb{Z}_p$ (or even the ring of integers of a finite extension of $\mathbb{Q}_p$), and $L/K$ finite then an integral basis exists.

I'm looking for a non-example ; $A$ Dedekind, $L/K$ finite for which there does not exist an integral basis. I suspect that taking $A$ to be the ring of algebraic integers of a finite extension of $\mathbb{Q}$ which is a UFD but not PID might do the trick.

Highly pathological examples (e.g. outside the realm of number fields) are very much welcome !