  We prove field quantifier elimination for valued fields endowed with both an
analytic structure and an automorphism that are $\sigma$-Henselian. From this
result we can deduce various Ax-Kochen-Ersov type results with respect to
completeness and the NIP property. The main example we are interested in is the
field of Witt vectors on the algebraic closure of $\mathbb{F}_{p}$ endowed with
its natural analytic structure and the lifting of the Frobenius. It turns out
we can give a (reasonable) axiomatization of its first order theory and that
this theory is NIP.
