Given a smooth projective curve $X/\mathbb{Q}_{p}$ of genus $g \ge 2$, there is an induced monodromy action of the Galois group on the $\ell$-adic cohomology of

$$X \otimes \bar{\mathbb{Q_{p}}}.$$

What is an explicit example where the action of wild inertia is non-trivial?

By ``explicit," I meant I would like an example that is not, say, a modular curve. Ideally, $X$ would be, say, presented as an effective divisor on smooth surface. Similarly, I would like to see the monodromy described without using automorphic methods.

(Here $\ell \ne p$ is a prime.)