Let $A$ be a commutative ring, consider the map $Spec(A[[t]])\rightarrow Spec(A)$, does it have geometrically connected fibers?
If $A$ is noetherian, it is clear because one has for $k$ a residue field of $A$:
$A[[t]]\otimes_{A}k=k[[t]]$
But such equality does not hold true in general. So what can we expect?