Examples of NIP fields of characteristic $p$ 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.
 A: For any field $K$ and ordered group $\Gamma$, the Malc'ev-Neumann field $K((\Gamma))$ is maximal by a Theorem of Krull, hence algebraically maximal. 
Taking a perfect infinite NIP field $K$, for instance $\mathbf F_p^{alg}$ and a non-divisible $p$-divisible ordered group $\Gamma$, for instance the subgroup $\langle \frac{1}{p^i}:i\in\mathbf N\rangle\subset\mathbf R^+$ provides an example of a non separably closed infinite NIP field.
A: Understanding concretely which fields (in the "pure" language of rings, $\mathcal{L} = \{0, 1, +, \cdot\}$) are NIP is a topic of current interest in model theory. The 2015 paper "Dp-minimal valued fields" by Jahnke, Simon, and Walsberg begins:
"Very little is known about NIP fields. It is widely believed that an NIP field is either real closed, separably closed or admits a definable henselian valuation. Note that even the stable case of this conjecture is open."
(http://arxiv.org/pdf/1507.03911v1.pdf)
So conjecturally, you may have to become familiar with a little valuation theory even to understand some interesting concrete examples.
To complement the 2/6/16 answer by Drike, we do have a fairly good idea of which fields are dp-minimal, where dp-minimal theories are a subclass of NIP theories in which one-variable definable sets are "indecomposable" in a certain sense (they generalize strongly minimal stable theories). A remarkable characterization of dp-minimal theories was given in 2015 by Will Johnson, and the answer involves a nontrivial amount of valued field theory:
"On dp-minimal fields," Will Johnson, http://arxiv.org/pdf/1507.02745v1.pdf
From that paper: "[the main theorem] almost says that all dp-minimal fields are elementarily equivalent to ones of the form $K((t^\Gamma))$ [AKA Malc'ev-Neumann fields] where $K$ is $\mathbb{F}^{alg}_p$ or a characteristic $0$ local field, and $\Gamma$ satisfies some divisibility conditions. The one exceptional case is the mixed characteristic case, which includes fields such as the spherical completion of $\mathbb{Z}^{un}_p(p^{1/{p^\infty}})$.''
