Let $G$ be a connected reductive group over a non archimedean local field $k$.
Let $P \subset G$ be a parabolic subgroup with Levi decomposition $P=MN$, $Z\_M \subset L$ be the center of $M$ and $S\_M \subset Z\_M$ be the maximal split torus.


It is commonly used that there exist element $s \in S\_M(k)$ whose action by conjugation on $N(k)$ is strictly contracting, meaning that for any compact open subgroup $N\_0 \subset N(k)$, the intersection $\bigcap\_{n \geq 0} s^n N\_0 s^{-n}$ is trivial.


This is equivalent to $v(\alpha(s))>0$ for all root $\alpha$ appearing in $\rm{Lie}(N)$, $v$ being the valuation on $k$.


Unfortunately I could not find a reference for this fact.



Strictly contracting elements in the center of a Levi subgroup
Let $G$ be a connected reductive group of an arbitrary field $k$, $P$ a parabolic $k$-subgroup, and $U = \mathscr{R}\_u(P)$ the unipotent radical of $P$ (so $U$ is what is unfortunately traditionally denoted as $N$). 


For any Levi $k$-subgroup $M$ of $P$ there is a 1-parameter $k$-subgroup $\lambda:{\rm{GL}}\_1 \rightarrow G$ such that $P = P\_G(\lambda)$, $M = Z\_G(\lambda)$, and
$U = U\_G(\lambda)$. In particular, $\lambda$ factors through the maximal split central $k$-torus $S\_M$ in $M$, so $\lambda(t) \in S\_M(k)$ for any $t \in k^{\times}$. By the Borel-Tits structure theory, if $S$ is a maximal split $k$-torus in $M$ containing $S\_M$ then it is also maximal as such in $G$ and the set $\Phi(G,S)$ of nontrivial $S$-weights on ${\rm{Lie}}(G)$ is a root system (spanning ${\rm{X}}(S)\_{\mathbf{Q}}$), possibly non-reduced.


The $k$-group $U$ is directly spanned in any order by the root groups $U\_a$ for non-multipliable $a \in \Phi(G,S)$ such that $\langle a, \lambda \rangle > 0$. (Of course, $\dim U\_a$ might be very large, and $U\_a$ might be non-commutative if $a$ is divisible.)


If $a$ is not divisible in $\Phi(G,S)$ then $U\_a$ is a vector group admitting a linear structure relative to which the effect of $\lambda(t)$-conjugation on $U\_a$ is scaling by $t^{\langle a, \lambda\rangle}$, and is an extension of such by another using $2a$ when $a$ is divisible. In this way we see that any $t \in k^{\times}$ with $|t| < 1$ yields $s := \lambda(t) \in S\_M(k)$ that does the job (as $\langle a, \lambda \rangle > 0$ precisely for those $a$ occurring as $S$-weights on ${\rm{Lie}}(U)$). 


