  We develop a characterisation of non-Archimedean derived analytic geometry
based on dg enhancements of dagger algebras. This allows us to formulate
derived analytic moduli functors for many types of pro-\'etale sheaves, and to
construct shifted symplectic structures on them by transgression using
arithmetic duality theorems. In order to handle duality functors involving Tate
twists, we introduce weighted shifted symplectic structures on formal weighted
moduli stacks, with the usual moduli stacks given by taking
$\mathbb{G}_m$-invariants.
  In particular, this establishes the existence of shifted symplectic and
Lagrangian structures on derived moduli stacks of $\ell$-adic constructible
complexes on smooth varieties via Poincar\'e duality, and on derived moduli
stacks of $\ell$-adic Galois representations via Tate and Poitou--Tate duality;
the latter proves a conjecture of Minhyong Kim.
