Artin-Schreier theory gives a bijection between the set of all degree-$p$ cyclic extension of a field $K$ of prime characteristic $p$ and the set of all $\mathbf{F}_p$-lines in the $\mathbf{F}_p$-space $K^+/\wp(K^+)$, where $\wp$ is the endomorphism $x\mapsto x^p-x$ of the additive group $K^+$ (thought of as an $\mathbf{F}_p$-space).
When $K=k((T))$ for some finite extension $k|\mathbf{F}_p$ of degree $f=[k:\mathbf{F}_p]$, the space $K^+/\wp(K^+)$ can be explicitly computed, along with its filtration. This is done for example in arXiv:0909.2541, form which the following is extracted.
Denote by $\frak{o}$ the ring of integers of $K$, and by $\frak{p}\subset\frak{o}$ the unique maximal ideal of $\frak{o}$. Concretely, we have ${\frak{o}}=k[[T]]$ and ${\frak{p}}=T{\frak{o}}$.
The filtration on the additive group $K^+$ is given by the powers $({\frak{p}}^n)$ of $\frak{p}$, indexed by $n\in\mathbf{Z}$. It induces a filtration on the $\mathbf{F}_p$-space $\overline{K}=K^+/\wp(K^+)$. We denote the induced filtration by $\overline{{\frak{p}}^n}$; we have $\overline{\frak{p}}=\{0\}$, and the codimension at each step is given by $$ \{0\} \subset_1\overline{\frak{o}} \subset_f\overline{\frak{p}^{-1}} \cdots \subset_f\overline{{\frak{p}}^{-pi+1}} =\overline{{\frak{p}}^{-pi}} \subset_f\cdots \quad\subset\overline{K^+} $$ Here, $i$ is any integer $>0$, and an inclusion of $\mathbf{F}_p$-spaces $E\subset_rE'$ means that $E$ is a codimension-$r$ subspace of $E'$.
The unramified degree-$p$ extension of $K$ corresponds to the $\mathbf{F}_p$-line $\overline{\frak{o}}$ (which can be canonically identified with $\mathbf{F}_p$ using the trace map $k\to\mathbf{F}_p$).
So every other line in this infinite-dimensional $\mathbf{F}_p$-space corresponds to a ramified cyclic degree-$p$ extension of $K$.