Let $\scr{G}$ be the absolute Galois group of $\mathbb{Q}_p$. Let $V$ be a $n$-dimensional $\mathbb{Q}_p$ vector space where $\scr{G}$ acts. Consider the representation $$\rho:\scr{G} \rightarrow \mathrm{GL}_n(\mathbb{Q}_p).$$
I have tried to show the following statement:
$\textit{If $\rho$ is irreducible then there exist only a finite number of stable lattices in $V$ (up to homothety)}$.
However, each of my attempts have miserably failed.
Could ask you an hint?
My first idea was: let $\Lambda_1,\dots,\Lambda_r$ be the stable lattices under the action of $\scr{G}$. Assume there exist a subrepresentation $W$ of $V$. By the theory I know that exist $\Lambda_W$ stable under the action of $\scr{G}$. However, I am not able to show that if I complete $\Lambda_W$ to a rank $n$ lattice then I find a lattice different of the $\Lambda_i$'s stable. (and probably this way is totally wrong)