Prove or disprove that every finite extension of $\text{Frac}\left(\Bbb{F}_{121}[X,Y]/(Y^2-f)\right)$ is separable

Question

Let $f\in\Bbb{F}_{121}[X]$ be irreducible, $R=\Bbb{F}_{121}[X,Y]/(Y^2-f)$ and $K=\text{Frac}(R)$. I know that every finite extension of finite fields is separable, but I couldn't show that $K$ is a finite field. I also tried to show that the Frobenius $F:x\mapsto x^{121}$ is an automorphism of $K$, which would imply $K$ is perfect, but I couldn't show that either.