Let $\mathbb{Z}_p$ be the $p$-adic integers. We have that the $p$-th cyclotomic polynomial is irreducible over $\mathbb{Z}_p$ applying the Eisenstein criterion (which is valid over $\mathbb{Z}_p$ when $p$ is the prime being used to prove irreducibility). So for $\omega$ a primitive $p$-th root of unity we have a ring extension $\mathbb{Z}_p[\omega]$ of degree $p-1$.
Is this ring also profinite? My attempt to prove it would be by considering extensions of the finite quotients $\mathbb{Z}/p^k\mathbb{Z}$ of $\mathbb{Z}_p$, but the $p$-th cyclotomic polynomial is equivalent to $(X-1)^{p-1}$ in $\mathbb{Z}/p\mathbb{Z}[X]$, so this extension would be trivial.
Is this the right way to go? Maybe it is irreducible (or at least doesn't have a root) modulo $p^2$ or $p^k$ for some $k$, but I'm low on ideas on how to analyse that.
Another way to go would be if the topology on $\mathbb{Z}_p[\omega]$ is the product topology. Therefore, as it is the product of profinite spaces, it must be again profinite. I'm actually unsure if there is a canonical way of extending the topology of a topological ring $R$ to any of it's finite extensions other than the product topology, or if there is a canonical topology of $\overline{\mathbb{Q}_p}$ that may not coincide with the product, so I don't know if this argument applies.