We introduce $p$-adic Kummer spaces of continuous functions on
$\mathbb{Z}_p$, that satisfy certain Kummer type congruences. We will classify
these spaces and show their properties, for instance, ring properties and
certain decompositions. As a result, these functions have always a fixed point
in $\mathbb{Z}_p$. A subclass of these functions has always a unique simple
zero in $\mathbb{Z}_p$. The fixed points and the zeros are effectively
computable by given algorithms. This theory can be transferred to values of
Dirichlet $L$-functions at negative integer arguments.