Non-vanishing of the structure-sheaf cohomology of affine henselian schemes and henselian affinoids

Let $(A,I)$ be a henselian pair such that $A$ is a domain of characteristic zero and $I\mathrm{\ne }0$. We prove that any flat affine henselian scheme of finite presentation over $(A,I)$ has nonzero first structure-sheaf cohomology whenever one of its fibers over a point in $V(I)$ has positive dimension. This in particular gives a negative answer to the question by Devadas in the mixed-characteristic case. As an application, we also prove that the first structure-sheaf cohomology does not vanish for henselian affinoids of positive dimensions.