I think I can give a characterization of your limit as a sum of Teichmüller representatives.
Let $q = p^f$ be some power of $p$. Let $Z_q = W(F_q)$ be the valuation ring of the unramified extension of $Q_p$ of degree $f$. Then for any $a$ in $Z_q$, there is a unique root of $x^q - x$ in $Z_q$ congruent to $a$ mod $p$. One can identify this with the limit, as n tends to infinity, of $a^{q^n}$.
I've never seen this before, but I guess you can do the same thing even if your extension is ramified. Let $R$ be some finite extension of $Z_p$. Let $F_q$ denote its residue field. Then for any $a$ in $R$, there is a unique root of $x^q - x$ in $R$ congruent to $a$ mod $p^{1/e}$, where $e$ is the ramification index. Again, it can be identified with the limit of $a^{q^n}$.
Assuming the limit you mentioned exists, it is the same as the limit of $a_1^{q^n} + \cdots + a_k^{q^n}$. And then this limit is the sum of the Teichmüller representatives that I just described.
