A space of p-adic test functions

In non-Archimedean distribution theory the space of Schwartz-Bruhat functions is often used as the space of test functions. This space has many nice properties, but it also has a serious shortcoming: it is not invariant under fractional derivatives. We present a space of test functions which has all the good properties of the Schwartz-Bruhat functions, but which in addition also is invariant under fractional derivatives. It is a Frechet space in a natural topology, and contains the Schwartz-Bruhat functions as a proper dense subspace.