  We prove a number of results on the \'etale cohomology of rigid analytic
varieties over $p$-adic non-archimedean local fields. Among other things, we
establish bounds for Frobenius eigenvalues, show a strong version of
Grothendieck's local monodromy theorem, prove mixedness of the nearby cycle
sheaf, and show that for any formal model, the IC sheaf on the special fiber is
captured by the nearby cycles of the IC sheaf on the generic fiber. We also
prove a local version of Deligne's weight-monodromy conjecture, by a novel
perfectoid analysis of nearby cycles.
  Along the way, we develop the theory of "constructible $\ell$-adic complexes
on Deligne's topos" (six operations, perverse t-structure, a notion of
mixedness, etc.), which is prerequisite to a precise discussion of the Galois
action on nearby cycles for algebraic and rigid analytic varieties over
non-archimedean fields.
