Does anyone know an example of a curve $X$ over a perfect field $k$ such that if $\tilde{X}$ is its noramlisation, there exists a point $x \in X$ and a point $y \in \tilde{X}$ over $x$ such that $k(y) / k(x)$ is not trivial?

(If we remove the hypothesis that it is over a field, there is such an example: $\mathbb{Z}[11\sqrt{3}] \to \mathbb{Z}[\sqrt{3}]$ and the points $(11, 11\sqrt{3}), (11)$.