We know that the fraction field $F:=\operatorname{Frac}(\mathbb Z_p[[X]])$ is strictly contained in the field of Laurent power series $\mathbb Q_p((X))$, thanks to this result of Gilmer. So my question is:
Is it possible to describe explicitly the elements of $F$?
Some similar questions have been already asked here or on Mathoverflow. Maybe the most relevant is this one regarding the explicit computation of the fraction field of $\mathbb Z[[X]]$. Someone suggests in the comments of the linked question that the problem with $\mathbb Z_p$ (instead of $\mathbb Z$) should be easier.
Some general necessary conditions are given here when the coefficients of the power series lie in any domain, but I'd like to find some sufficient conditions in the particular case of $\mathbb Z_p$.
Many thanks in advance