Nonarchimedean bivariant K-theory

. We introduce bivariant K -theory for nonarchimedean bornological algebras over a complete discrete valuation ring V . This is the universal target for dagger homotopy invariant, matricially stable and excisive functors, similar to bivariant K -theory for locally convex topological C -algebras and algebraic bivariant K -theory. As in the archimedean case, we use the universal property to construct a bivariant Chern character into analytic and periodic cyclic homology. When the first variable is the ground algebra V , we get a version of Weibel’s homotopy algebraic K -theory, which we call stabilised overconvergent analytic K -theory . The resulting analytic K -theory satisfies dagger homotopy invariance, stability by completed matrix algebras, and excision.