  We formulate and study a torsion analogue of the weight-monodromy conjecture
for a proper smooth scheme over a non-archimedean local field. We prove it for
proper smooth schemes over equal characteristic non-archimedean local fields,
abelian varieties, surfaces, varieties uniformized by Drinfeld upper half
spaces, and set-theoretic complete intersections in toric varieties. In the
equal characteristic case, our methods rely on an ultraproduct variant of Weil
II established by Cadoret.
