  Let $k$ be a field with a nontrivial discrete valuation which is complete and
has perfect residue field. Let $G$ be the group of $k$-rational points of a
reductive, linear algebraic group $\textbf{G}$ equipped with an involution
$\theta$ defined over $k.$ Let $\mathfrak{p}$ denote the $(-1)$-eigenspace in
the decomposition of the Lie algebra of $G$ under the differential $d\theta.$
If $\textbf{H}$ is a subgroup of $\textbf{G}^{\theta}$, the set of
$\theta$-fixed points, which contains the connected component of
$\textbf{G}^{\theta},$ then $H=\textbf{H}(k)$ acts on $\mathfrak{p}$, which we
treat as a symmetric space. Let $r \in \mathbb{R}.$ Under mild restrictions on
$\textbf{G}$ and $k,$ the set of nilpotent $H$-orbits in $\mathfrak{p}$ is
parametrized by equivalence classes of noticed Moy-Prasad cosets of depth $r$
which lie in $\mathfrak{p}.$
