Toward an ultrametric calculus in a field K with an infinite rank valuation.

K is the scalar field of the first orthomodular (or Form Hilbert) space, described by H. Keller in 1980. It has a non-archimedean order, an infinite rank valuation as well as an explicitly defined ultrametric, all of which induce the same topology. We will present results concerning analytic functions, and show some examples that sharply contrast with the case of rank one valuations. Finally we prove a theorem concerning local invertibility for C(1) functions.