  Let $R$ be a polynomial or formal power series ring with coefficients in a
DVR $V$ of mixed characteristic with a uniformizer $\pi$. We prove that the
$R$-module annihilator of any nonzero $\D(R,V)$-module is either zero or is
generated by a power of $\pi$. In contrast to the equicharacteristic case,
nonzero annihilators can occur; we give an example of a top local cohomology
module of the ring $\mathbb{Z}_2[[x_0, \ldots, x_5]]$ that is annihilated by
$2$, thereby answering a question of Hochster in the negative.
