Question I want to ask is practically precisely what's in the question but I will restate to make it clearer.
Suppose $k$ is a field of characteristic $p$ which is not algebraically closed. Then we know that $k$ has an infinite absolute galois group by Artin-Schreier. But does the absolute galois group $G$ of $k$ necessarily contain $\mathbb{Z}_p$ if the $p$-part of $G$ is not trivial?
Now I believe this to be true by generalisation of Artin-Schreier (different one to the one I have just mentioned) by Witt. This basically says as long as there exists $x\in k$ such that $x$ is not of the form $\mathscr{y}=y^p-y$ for some $y\in k$, then there exists a cyclic extension of degree $p^n$ for all $n$ which would prove my statement. However I am currently not sure whether this argument really works and I'm trying to think of a case where $k=\mathscr{P}k$.
The analogue of Kummer's theory for a field $k$ of characteristic $p$ is the Artin-Schreier-Witt theory. It allows in particular to describe the Galois group $G_p$ of the maximal abelian pro-$p$-extension of $k$ in terms of the ring of Witt vectors $W(k)$ and the Artin-Schreier operator $P$. More precisely, $G_p$ is topologically isomorphic to $Hom(W(k)/P(W(k)), W(F_p))$. Noting that $W(F_p)$ is topologically isomorphic to $Z_p$, you get all the $Z_p$-extensions of $k$ .
Unfortunately I know of no book form account of the ASW theory. Perhaps the best introduction for beginners is the first chapter of Lara Thomas' thesis "Arithmétique des extensions d'ASW " , Univ. Toulouse Le Mirail .