Elliptic curve which attains potential good reduction over an Artin-Schreier extension.

Question

I am looking for an elliptic curve $E$ over the field $\overline{\mathbb{F}_{p}}((t))$, which attains good reduction over an Artin-Schreier extension of $\overline{\mathbb{F}_{p}}((t))$, i.e.: an extension of the form $\overline{\mathbb{F}_{p}}((t))[x]/(x^{p}-x-y)$, with $y \notin \{x^p-x| x \in \overline{\mathbb{F}_{p}}((t))\}$. Here, $\overline{\mathbb{F}_{p}}$ denotes a fixed algebraic closure of $\mathbb{F}_{p}$, and $\overline{\mathbb{F}_{p}}((t))$ denotes the field of formal Laurent series in the formal variable $t$. Or is there maybe any reason why such an elliptic curve cannot exist?