Let $X \subset \mathbb{A}_{\mathbb{C}}^2$ be a closed subscheme, and let $\pi \colon X \to \mathbb{A}_{\mathbb{C}}^1$ be a finite flat map. For a point $p \in X(\mathbb{C})$, is it true that we can find an arbitrarily small open neighborhood $U \ni p$ (in the analytic topology) such that for some analytic-open neighborhood $V \ni \pi(p)$, the degree of the fiber $\pi^{-1}(q) \cap U$ is constant over $q \in V$?
What I know so far: I know that fiber degree is constant in finite flat families, but I'm interested in whether this property is analytic-local on the source in the above sense.