  A parametrised diffusion operator on the regular domain $\Omega$ of a
$p$-adic Schottky group is constructed. It is defined as an integral operator
on the complex-valued functions on $\Omega$ which are invariant under the
Schottky group $\Gamma$, where integration is against the measure defined by an
invariant regular differential 1-form $\omega$. It is proven that the space of
Schottky invariant $L^2$-functions on $\Omega$ outside the zeros of $\omega$
has an orthonormal basis consiting of $\Gamma$-invariant extensions of Kozyrev
wavelets which are eigenfunctions of the operator. The eigenvalues are
calculated, and it is shown that the heat equation for this operator provides a
unique solution for its Cauchy problem with Schottky-invariant continuous
initial conditions supportes outside the zero set of $\omega$, and gives rise
to a strong Markov process on the corresponding orbit space for the Schottky
group whose paths are c\`adl\`ag.
