On uniqueness of $p$-adic meromorphic functions

Let $K$ be a complete ultrametric algebraically closed field of characteristic zero, and let ${\mathcal{M}} (K)$ be the field of meromorphic functions in $K$. For all set $S$ in $K$ and for all $f\in {\mathcal{M}}(K)$ we denote by $E(f,S)$ the subset of $K {\times } {\mathbb{N}}^{*}$: ${\bigcup _{ a\in S}}\{(z,q)\in K {\times } \mathbb{N}^{*} \vert z$ zero of order $q \text{ of} f(z)-a\}.$ After studying unique range sets for entire functions in $K$ in a previous article, here we consider a similar problem for meromorphic functions by showing, in particular, that, for every $n\geq 5$, there exist sets $S$ of $n$ elements in $K$ such that, if $f, g\in {\mathcal{M}} (K)$ have the same poles (counting multiplicities), and satisfy $E(f,S)=E(g,S)$, then $f=g$. We show how to construct such sets.