  In this work, we analyze the finite element method with arbitrary but fixed
polynomial degree for the nonlinear Helmholtz equation with impedance boundary
conditions. We show well-posedness and error estimates of the finite element
solution under a resolution condition between the wave number $k$, the mesh
size $h$ and the polynomial degree $p$ of the form ``$k(kh)^{p+1}$ sufficiently
small'' and a so-called smallness of the data assumption. For the latter, we
prove that the logarithmic dependence in $h$ from the case $p=1$ in [H.~Wu,
J.~Zou, \emph{SIAM J.~Numer.~Anal.} 56(3): 1338-1359, 2018] can be removed for
$p\geq 2$. We show convergence of two different fixed-point iteration schemes.
Numerical experiments illustrate our theoretical results and compare the
robustness of the iteration schemes with respect to the size of the
nonlinearity and the right-hand side data.
