  Let $M$ be chosen uniformly at random w.r.t. the Haar measure on the unitary
group $U_n$, the unitary symplectic group $USp_{2n}$ or the orthogonal group
$O_n$. Diaconis and Shashahani proved that the traces
$\mathrm{tr}(M),\mathrm{tr}(M^2),\ldots,\mathrm{tr}(M^k)$ converge in
distribution to independent normal random variables as $k$ is fixed and
$n\to\infty$. Recently, Gorodetsky and Rodgers proved analogs for these results
for matrices chosen from certain finite matrix groups. For example, let $M$ be
chosen uniformly at random from $U_n(\mathbb{F}_q)$. They show that
$\{\mathrm{tr}(M^i)\}_{i=1,p\nmid i}^{k}$ converge in distribution to
independent uniform random variables in $\mathbb{F}_{q^2}$ as $k$ is fixed and
$n\to\infty$.
  We prove analogs for these results over local fields. Let $\mathcal{F}$ be a
local field with a ring of integers $\mathcal{O}$, a uniformizer $\pi$, and a
residue field of odd characteristic. Let $\mathcal{K}/\mathcal{F}$ be an
unramified extension of degree $2$ with a ring of integers $\mathcal{R}$. Let
$M$ be chosen uniformly at random w.r.t. the Haar measure on the unitary group
$U_n(\mathcal{O})$, and fix $k$. We prove that the traces of powers
$\{\mathrm{tr}(M^i)\}_{i=1,p\nmid i}^k$ converge to independent uniform random
variables on $\mathcal{R}$, as $n\to\infty$. We also consider the case where
$k$ may tend to infinity with $n$. We show that for some constant $c$ (coming
from the mod $\pi$ distribution), the total variation distance from independent
uniform random variables on $\mathcal{R}$ is $o(1)$ as $n\to\infty$, as long as
$k<c\cdot n$. We also consider other matrix groups over local fields and prove
similar results for them. Moreover, we consider traces of powers $M^{pi}$ and
traces of negative powers, and show that apart from certain necessary modular
restrictions, they also equidistribute in the limit.
