Here is a beautiful and essentially elementary result using the $p$-adics: the Skolem-Mahler-Lech theorem.
Theorem. (Skolem-Mahler-Lech) Let $(a_i)$ be a sequence defined by an integer linear recurrence. Then the set of $i$ such that $a_i=0$ is the union of a finite set with finitely many arithmetic progressions.
A quick proof may be found on Terry Tao's blog, here. Essentially, the $p$-adic step of the proof works by defining a $p$-adic analytic function with infinitely many zeros, and then concluding that this function is identically zero--by the definition of this function, this gives some congruence information about the structure of the zero set of the linear recurrence, as desired. The proof is quite elementary and beautiful, and I think accessible to people seeing the $p$-adics for the first time.
