Relative Poincar\'e duality in nonarchimedean geometry

We prove a conjecture of Bhatt-Hansen that derived pushforwards along proper morphisms of rigid-analytic spaces commute with Verdier duality on Zariski-constructible complexes. In particular, this yields duality statements for the intersection cohomology of proper rigid-analytic spaces. In our argument, we construct cycle classes in analytic geometry as well as trace maps for morphisms that are either smooth or proper or finite flat, with appropriate coefficients. As an application of our methods, we obtain new, significantly simplified proofs of $p$-adic Poincar\'e duality and the preservation of $\mathbf{F}_p$-local systems under smooth proper higher direct images.