Let $A\rightarrow B$ be a local morphism of complete noetherian rings making $B$ a formally smooth $A$-algebra. Does the induced morphism $\textrm{Spec}(B)\to\textrm{Spec}(A)$ have geometrically regular fibers ?

Noting $k$ the residue field of $A$, the answer is yes if $k$ is of characteristic $0$, by resolution of singularities. If $K$ is the residue field of $B$, the answer is also yes if the extension $K/k$ is finite by EGA IV 2, paragraph 7, theorem (7.5.1). The result subsists if the extension is of finite type, even if cannot find reference for this, but does the result subsist if $B\otimes_A k$ is the separable closure of $k$ ?