Here is a finite-ness result for $p$-adic fields. Let $k$ be a p-adic field. One knows that there are only finitely many square classes of $k$ -- that is $k^{\times} / k^{\times 2}$ is finite (as a set). Thus the map $k \rightarrow k$ given by $x \mapsto x^2$ misses only finitely many points. I think a A similar situation holds should hold for $p$-th classes p^{th}$-classes of $l$-adic fields. As a concrete example, we know that for $k = \mathbb{Q}_2$, the square classes $k^\times/k^{\times 2}$ are generated by $2,-1,5$ (I think), 2,-1,5$, in particular, $k^{\times} / k^{\times 2} \cong (\mathbb{Z}/2)^{\times 3}$.
This doesn't answer the question at all, but its a finitely-generated analogue of the finite-ness result you want.
