Fréchet spaces of non-archimedean valued continuous functions

Let X be an ultraregular space and let K be a complete non-archimedean non-trivially valued field. Assume that the locally convex space E = C c ( X ; K ) of all continuous functions from X to K with the topology τ c of uniform convergence on compact subsets of X is a Fréchet space. We shall prove that E has an orthogonal basis consisting of K -valued characteristic functions of clopen (i.e. closed and open) subsets of X and that it is isomorphic to the product of a countable family of Banach spaces with an orthonormal basis.