Let $K$ be the set of open-closed subsets of $\mathbb{Z}_p$. Let $M$ be the set of functions from $K$ to $C_p$ that are additive under disjoint unions. Then $M$ can be regarded as an elementary abelian pro-$p$ group: multiplication is pointwise in $C_p$, and a base of open subgroups is given by $\{ U_n \}$, where $U_n$ consists of those functions which map cosets of $p^n \mathbb{Z}_p$ to $0$. Moreover, $\mathbb{Z}_p$ has a translation action on $K$, and hence a continuous action on $M$ in which each of the $U_n$ is normalised. We use this action to construct a group $G = M \rtimes \mathbb{Z}_p$. This is realisable as an inverse limit of the groups $C_p \wr C_{p^n}$ (regular wreath product); in particular, it is a $2$-generator pro-$p$ group.

Now for the question: Is the group $G$ above isomorphic to a group arising from some standard construction, and if so does it have a nice name? I would be surprised if nobody has used this group before as an example of something or other. Also, there seems to be quite a general construction behind this.

Edit: the group described is a subgroup of the pro-$p$ group $C_p \wr_K \mathbb{Z}_p$. The unusual part is the extra condition that the functions should be additive (motivation: I wanted $G$ to have relatively few normal subgroups). Are there alternative conditions one could impose that would result in an isomorphic group? In particular, I'm relying on the fact that $C_p$ is abelian, which is bad for generalisations.