  We prove a Poincar\'e duality for arithmetic $p$-adic pro-\'etale cohomology
of smooth dagger curves over finite extensions of ${\mathbf Q}_p$. We deduce
it, via the Hochschild-Serre spectral sequence, from geometric comparison
theorems combined with Tate and Serre dualities. The compatibility of all the
products involved is checked via reduction to the ghost circle, for which we
also prove a Poincar\'e duality (showing that it behaves like a proper smooth
analytic variety of dimension $1/2$). Along the way we study functional
analytic properties of arithmetic $p$-adic pro-\'etale cohomology and prove
that the usual cohomology is nuclear Fr\'echet and the compactly supported one
-- of compact type.
