  Over any smooth algebraic variety over a $p$-adic local field $k$, we
construct the de Rham comparison isomorphisms for the \'etale cohomology with
partial compact support of de Rham $\mathbb Z_p$-local systems, and show that
they are compatible with Poincar\'e duality and with the canonical morphisms
among such cohomology. We deduce these results from their analogues for rigid
analytic varieties that are Zariski open in some proper smooth rigid analytic
varieties over $k$. In particular, we prove finiteness of \'etale cohomology
with partial compact support of any $\mathbb Z_p$-local systems, and establish
the Poincar\'e duality for such cohomology after inverting $p$.
