I'm looking for a result that must be known: Do all non-separable curve morphisms over a field with characteristic $p>0$ have degree divisible by $p$? I couldn't find this exact result being stated, but it is heavily implied by this proposition in Silverman's Arithmetic of Elliptic Curves:
Every map $\psi: C_1 \rightarrow C_2$ of smooth curves over a field of characteristic $p > 0$ factors as $$C_1 \xrightarrow{\quad\phi\quad} C_1^{(q)} \xrightarrow{\quad\lambda\quad} C_2,$$ where $q = \deg_i(\psi)$, the map $\phi$ is the $q^{\text{th}}$-power Frobenius map, and the map $\lambda$ is separable.
The proposition assumes that the inseparable degree is a $q=p^n$, a power of the field characteristic. I'm assuming there is a result that says so.