In Milne Prop 2.29, it is said that the integral closure $B$ of a PID $A$ in a separable finite extension of its fraction field is a free $A$-module. On the other hand, I have read here that if the base ring is a complete DVR, $\mathrm{Frac}(B)$ need not be separable over $\mathrm{Frac}(A)$ for $B$ to be finitely generated over $A$ (although I would very much like to see a reference for this), but my question is : is it still a free $A$-module ?
The question in the title is a little more restrictive (although not much), but is what I'm really interested in.