Does Tate's $p$-adic uniformisation theorem hold over general non-archimedean local fields?

Tate's $p$-adic uniformisation theorem for elliptic curves goes as follows:

Let $K$ be a $p$-adic field, let $E/K$ be an elliptic curve with $v_K(j) \ge 0$, and let $\gamma(E/K)=-c_4/c_6 \in K^{\times}/(K^{\times})^2$.

a) There is a unique $q \in K^{\times}$ with $|q|<1$ such that $E$ is isomorphic over $\overline{K}$ to the Tate curve $E_q$. Further $q \in K$.

b) The following 3 conditions are equivalent:

i) $E$ is isomorphic to $E_q$ over $K$

ii) $\gamma(E/K)=1$.

iii) $E$ has split multiplicative reduction.

Silverman's textbook and everyone else seems to state the theorem with the assumption that $K$ is a $p$-adic field. I know that the definition of the Tate cuve and the isomorphism between $K^{\times}/q^{\mathbb{Z}} \cong E(K)$ holds over any complete non-archimedean local field (Silverman mentions this too).

I want to know if the theorem I state above holds in more generality than just $p$-adic fields or if it is restricted to only $p$-adic fields. If it is true in general, I'd be grateful if someone can attach a reference. Thank you so much!