Consider the power series $f(x)=\sqrt{\frac{1-x}{1-x^p}}$ over the algebraic closure of $\mathbb{Q}_p$, defined by $f(0)=1$. What can be said about an analytic continuation "in the form of Mittag-Leffler"? By that I mean that $f(x)$ can be expressed as $f(x)=g(x)+\sum_{n=0}^{\infty} \frac{a_n(x)}{h(x)^n}$, where $g(x)$ converges in a disk greater than the unit ball, $h(x)$ is a polynomial of norm $\le 1$, and the $a_n(x)$ are polynomials satisfying $\text{ord}_p(a_n)\ge rn-m$ for some $r>0$ and a constant $m \in \mathbb{R}$. Here the order of a polynomial is the minimum order of its coefficients.

I am particuarly interested in the values $f(\lambda)$, for Teichmuller representatives $\lambda$, which satisfy $\lambda^q=\lambda$ for some $q=p^r$. Thus I would like $h(x)$ not to have a zero at the Teichmuller representatives. Apparently the values at the Teichmuller representatives are related to classical results on Gauss Sums but I couldn't find anything in the literature.