  Let $V$ be a complete discrete valuation ring with fraction field $F$ of
characteristic zero and with residue field $\mathbb{F}$. We introduce analytic
cyclic homology of complete torsion-free bornological algebras over $V$. We
prove that it is homotopy invariant, stable, invariant under certain nilpotent
extensions, and satisfies excision. We use these properties to compute it for
tensor products with dagger completions of Leavitt path algebras. If $R$ is a
smooth commutative $V$-algebra of relative dimension $1$, then we identify its
analytic cyclic homology with Berthelot's rigid cohomology of
$R\otimes_{V}\mathbb{F}$.
