  We introduce and develop $(\pi,p)$-adic Dwork theory for $L$-functions of
exponential sums associated to one-variable rational functions, interpolating
$p^k$-order exponential sums over affinoids. Namely, we prove a generalization
of the Dwork-Monsky-Reich trace formula and apply it to establish an analytic
continuation of the $C$-function $C_f(s,\pi)$. We compute the lower
$(\pi,p)$-adic bound, the Hodge polygon, for this $C$-function. Along the way,
we also show why a strictly $\pi$-adic theory will not work in this case.
