  For fixed prime integer $p > 0$ we develop a notion of Bernstein-Sato
polynomial for polynomials with $\mathbb{Z} / p^m$-coefficients, compatible
with existing theory in the case $m = 1$. We show that the ``roots" of such
polynomials are rational and we show that the negative roots agree with those
of the mod-$p$ reduction. We give examples to show that, surprisingly, roots
may be positive in this context. Moreover, our construction allows us to define
a notion of ``strength" for roots by measuring $p$-torsion, and we show that
``strong" roots give rise to roots in characteristic zero through mod-$p$
reduction.
