Yes, unless I'm missing some subtlety.
Theorem 4.20 of Mumford's paper reads:
Every stable curve over $S$ with nonsingular generic fiber and $k$-split degenerate closed fiber is isomorphic to $P_{\Gamma}$ for a unique$^\ast$ flat Schottky group $\Gamma \subset PGL(2,K)$.
Here $S = \mathrm{Spec}\ A$ where $A$ is a complete integrally closed noetherian local ring, $K = \mathrm{Frac}\ A$, and $k$ is the residue field of $A$. Saying a stable curve $X$ over $k$ is $k$-split degenerate means (see shortly after Defn 3.2)
The components of the normalization $X$ are all $\mathbb{P}^1_k$'s.
The nodes of $X$ are all $k$ points.
Any node locally looks like $k[x,y]/(xy)$. For example, working over $\mathbb{R}$, a node of the form $\mathbb{R}[x,y]/(x^2+y^2)$ is forbidden.
Conditions (1) and (2) are true in your setting, and condition (3) follows from the fact that your nodes always join two different components, not one component and itself.
$^*$ Mumford must mean "unique up to conjugacy".