Several approaches to non-archimedean geometry, in M. Baker et al., p-adic Geometry

Let k be a non-archimedean field: a field that is complete with respect to a specified nontrivial non-archimedean absolute value | · |. There is a classical theory of k-analytic manifolds (often used in the theory of algebraic groups with k a local field), and it rests upon versions of the inverse and implicit function theorems that can be proved for convergen