Let $R$ be a discrete valuation ring with fraction field $K$ and residue field $k$. Assume that the characteristic of $K$ is $0$ and of $k$ is $p>0$. Let $\pi:X \to \mbox{Spec}(R)$ be a flat, projective morphism with $X$ a regular scheme. Is it true that a general hypersurface of $H_X$ is a regular scheme, flat over $\mbox{Spec}(R)$? 

If I understand correctly, this is true "locally" by a result of Flenner proven in [Die Sätze von Bertini für lokale Ringe][1]. 


  [1]: https://link.springer.com/article/10.1007%2FBF01351596