A question on linear algebra over non-Archimedean local field Let $\mathbb{F}$ be a non-Archimedean local field. Let $\{T_a\}_{a=1}^\infty$ be a sequence of linear operators $\mathbb{F}^n\to\mathbb{F}^n$ of rank $n$.  After a choice of subsequence, is it possible to construct  sequences of vectors $\{e^i_a\}\subset \mathbb{F}^n$, $i=1,\dots, n,$ such that the following properties are satisfied:
(1) for any $a$ the vectors $e^1_a,\dots, e^n_a$ form a basis of $\mathbb{F}^n$ and when $a\to \infty$ they converge to another basis of $\mathbb{F}^n$;
(2)  the vectors $T_a(e^1_a),\dots, T_a(e^n_a)$ also form a basis of $\mathbb{F}^n$ and when $a\to \infty$ appropriate multiples of these vectors converge to another basis of $\mathbb{F}^n$?
Remark. In the Archimedean case the answer is positive. Indeed fix a Euclidean (resp. Hermitian) metric. $T_a$ admits an orthonormal  basis $e^1_a,\dots, e^n_a$ such that $T_a(e^1_a),\dots, T_a(e^n_a)$ are pairwise orthogonal vectors. After a choice of subsequence all the required properties will be satisfied.
 A: $\def\FF{\mathbb{F}}\def\GL{\text{GL}}$This is basically the same thing YCor sketches in his comment: Let $R$ be the ring of integers of $\FF$. Let $K$ be the group of invertible matrices with entries in $R$ whose inverses are also in $R$, and let $T$ be the group of diagonal matrices.
The ring $R$ is a dvr so, by the Smith normal form theorem, each of your matrices $T_a$ can be factored as $U_a D_a V_a$ with $D_a \in T$ and $U_a$, $V_a \in K$.
The group $K$ is compact in the non-archimedean topology. Proof: $K$ can be written as $\{ (g,h) : gh = \text{Id}_n,\ g,h \in \text{Mat}_{n \times n}(R) \}$. This is a closed subspace of $\text{Mat}_{n \times n}(R)^2$, and $\text{Mat}_{n \times n}(R)^2 \cong R^{2n^2}$ is obviously compact. So we can extract a subsequence where the $U_a$ approach a limit $U$ and the $V_a$ approach a limit $V$.
Then we take $(e_a^1, e_a^2, \ldots, e_a^n)$ to be the columns of $V_a^{-1}$, just as in the archimedean case.
I learned the slogan "Smith normal form is non-archimedean singular value decomposition" from Kiran Kedlaya.
