locally analytic representations were introduced by peter schneider and jeremy teitelbaum as a tool to understand $p$-adic langlands program. from the very beginning the theory of $p$-valued groups played an instrumental role in the study of locally analytic representations. in a previous work we attached a rigid analytic group to a  $\textit{$p$-saturated group}$ (a class of $p$-valued groups that contains uniform pro-$p$ groups and pro-$p$ iwahori subgroups as examples). in this talk i will outline how to elevate the rigid group to a $\textit{dagger group}$, a group object in the category of dagger spaces as introduced by elmar grosse-klönne. i will further introduce the space of $\textit{overconvergent functions}$ and its strong dual the $\textit{overconvergent distribution algebra}$ on such a group. finally i will show that in analogy to the locally analytic distribution algebra of compact $p$-adic groups, in the case of uniform pro-$p$ groups the overconvergent distribution algebra is a fr´echet-stein algebra, i.e., it is equipped with a desirable algebraic structure. this is joint work with claus sorensen and matthias strauch.