Definition. According to Shelah, a field $K$ does not have the independence property (i.e. is NIP) if for every first order formula $\varphi(x, \bar y)$ in the language of fields $(+,\times,0,1)$, the Vapnik–Chervonenkis dimension of the family of subsets$\{\varphi(K, \bar k) : \bar k \in K^n\}$ is finite, i.e. there do not exist arbitrarily large finite sets $A \subset K$ whose subsets are all of the form $A \cap \varphi(K, \bar k)$ for some $\bar k$ from $K^n$.
Examples of NIP fields include, in characteristic $p$ finite fields (easy), $\mathbf F^{alg}$, and in zero characteristic, $\mathbf C$, $\mathbf R$ and $\mathbf Q_p$.
A Theorem of I. Kaplan asserts that a NIP field has no Artin-Schreier extension. I found
Theorem (I. Kaplan, F. Wagner and T. Scanlon). A valued field of characteristic $p$ with perfect infinite NIP residue field, with $p$-divisible value group and which is algebraically maximal ($i.e.$ with no proper algebraic valued extension having both same residue field and same valued group) is NIP.
I am not familiar with valuation theory, so the above is of little help to me, but for those who know,
Question: would you have any concrete examples of NIP fields of characteristic $p>0$ ? (non separably closed)
I am looking for one which has a cyclic (Galois) extension.