  Let $G$ be a multiplicative subgroup of $\mathbb{Q}_p$. In this paper, we
describe the theory of the pair $(\mathbb{Q}_p, G)$ under the condition that
$G$ satisfies Mann property and is small as subset of a first-order structure.
First, we give an axiomatisation of the first-order theory of this structure.
This includes an axiomatisation of the theory of the group $G$ as valued group
(with the valuation induced on $G$ by the $p$-adic valuation). If the subgroups
$G^{[n]}$ of $G$ have finite index for all $n$, we describe the definable sets
in this theory and prove that it is NIP. Finally, we extend some of our results
to the subanalytic setting.
