Let $K$ be a finite extension of $Q_p$. Let $q$ be the size of the residue field of $K$, and let $\pi$ be a uniformizer of $K$. Then $q/\pi$ is some power of $\pi$ up to a unit $u$ in $K$, say $q/\pi = u\pi^n$. The question is does there exist a power series or even a Laurent series $f(X)$ with coefficients in $K$ such that $f$ takes $(q/\pi)^i$ to $\pi^i$ for each $i \ge 0$? If such $f$ exists then it is essentially acting as an $n$-th root function on certain values in the maximal ideal of $\mathcal{O}_K$.
If it is not the case that such $f$ exists then we can relax the conditions a little and ask a similar question: for each $N > 0$ does there exist $f_N(X)$ such that $f_N((q/\pi)^i) = \pi^i$ for all $i \le N$? A positive answer to this question would be just as useful to me as a positive answer to the first question.
I have tried to find such functions by taking $\text{exp}_p(\frac{1}{n}\text{log}_p(1+X))$, the only issue with doing this is that it only allows for taking $n$-th roots on principal units, while I need to take $n$-th roots of values in the maximal ideal.
