Example of integral extension of degree $p$?

Question

For a prime $p$, does there exist an example of a (pair of) domains (if yes any way to construct?) $R$ and $S$ such that

1. Both $R$ and $S$ are normal complete local domains
2. $S$ contains an alg. closed field of char 0.
3. $R\hookrightarrow S$ is an integral extension of degree $p$

and, if possible, I also would like to have:

4. $\mathbb{Z}_p$ acts on $S$ with $R=S^{\mathbb{Z}_p}$, the ring of invariants.

I was hoping to construct with something from $\mathbf{k}[[ X ]]$, but having no further thoughts.