  Let $(G,\omega)$ be a $p$-saturated group and $K/\mathbb{Q}_p$ a finite
extension. In this paper we introduce the space of $K$-valued overconvergent
functions $\mathcal{C}^\dagger(G,K)$. In the process we promote the rigid
analytic group attached to $(G,\omega)$ in a previous work of the first two
authors to a dagger group. A main result of this article is that under certain
assumptions (satisfied for example when $G$ is a uniform pro-$p$ group) the
distribution algebra $D^\dagger(G,K)$, i.e. the strong dual of
$\mathcal{C}^\dagger(G,K)$, is a Fr\'{e}chet-Stein algebra in the sense of
Schneider and Teitelbaum.
  In the last section we introduce overconvergent representations and show that
there is an anti-equivalence of categories between overconvergent
$G$-representations of compact type and continuous $D^\dagger(G, K)$-modules on
nuclear Fr\'{e}chet spaces. This is analogous to the anti-equivalence between
locally analytic representations and modules over the locally analytic
distribution algebra as proved by Schneider and Teitelbaum.
