Yes, unless I'm missing some subtlety.

Theorem 4.20 of [Mumford's paper][1] 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)

 1. The components of the normalization $X$ are all $\mathbb{P}^1_k$'s.

 2. The nodes of $X$ are all $k$ points.

 3. 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". 


  [1]: http://archive.numdam.org/ARCHIVE/CM/CM_1972__24_2/CM_1972__24_2_129_0/CM_1972__24_2_129_0.pdf