sequence $\{a^{p^{n}}\}$ converges in the p-adic numbers.
Recall that the Euler totient function has values $\phi(p^n)=p^{n-1}(p-1)=p^n-p^{n-1}$ for all $n$. This means that for all $a$ coprime to $p$ we have the congruence $$ a^{p^n}\equiv a^{p^{n-1}}\pmod{p^n}. $$ By raising that congruence to power $p^{m-n}$ a straightforward induction on $m$ proves that $$ a^{p^m}\equiv a^{p^{n-1}}\pmod{p^n} $$ for all $m\ge n$. This holds for all $n$ implying that the sequence is Cauchy, and thus convergent w.r.t. the $p$-adic metric.