This won't qualify as something you can explain to undergraduate students, but non-archimedean dynamics has recently seen a number of applications to classical complex dynamics. (Non-archimedean is dynamics over a field with a non-archimedean absolute value, but not specifically an extension of $\mathbb{Q}_p$.) I'll mention one beautiful example, which is a recent theorem of Matt Baker and Laura DeMarco. Let
$$f_c(x) = x^2+c$$
be the usual quadratic polynomial, and for any starting value $a$, let $O_c(a)$ be the forward orbit of $a$ for the map $f_c$. That is,
$$O_c(a) = \{a,f_c(a),f_c^2(a),f_c^3(a),...\}$$
where $f_c^n$ denotes the $n$'th iterate of $f_c$.
Theorem: Let $a$ and $b$ be complex numbers with $a^2\ne b^2$. Then
$$\{c\in\mathbb{C} : O_c(a) \text{ and } O_c(b) \text{ are both finite}\}$$
is a finite set.
The proof is partly complex dynamics, partly equidistribution theorems (in both the complex and $p$-adic settings), and partly a reduction step in which one works in Berkovich space over a non-archimedean field. Note that the statement of the theorem is purely a statement about complex numbers, but the proof requires non-trivial methods from non-archimedean analysis.
