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][1], 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 $K$.

  [1]: http://arxiv.org/abs/0909.2541