Consider $k$ a finite extension of $\mathbb {Q}_p$, with $p$ a prime number. Let $H$ be a finite index subgroup of $k^*$ and $G$ be the group $\mathrm {SL}(n,k)$ with its Zariski topology of $\mathbb {Q} _p$-group. We investigate the existence of a subgroup of $G$ which is Zariski-dense and such that each of its elements has a spectrum included in $H$. A necessary and sufficient condition is obtained : such a subgroup exists if and only if either $-1$ belongs to $H$ or the dimension n is not congruent to 2 modulo 4.