For a research project, I'd like to have the following result:
Let $f \in \mathbb{Z}_p[x]$ be a polynomial of degree $d$, and $r$ be a positive integer. Then the set of $a \in \mathbb{Z}_p$ such that $f(a) \equiv 0 \mod p^r$ is the union of at most $d$ disks in $\mathbb{Z}_p$.
I was able to cobble together a proof by induction on $r$, but the result is so elegant that I can't be the first person to consider it. Does it crop up in standard texts, such as in the development of $p$-adic analysis or Berkovich spaces?