  We calculate the nucleon axial form factor up to the leading one-loop order
in a covariant chiral effective field theory with the $\Delta(1232)$ resonance
as an explicit degree of freedom. We fit the axial form factor to the latest
lattice QCD data and pin down the relevant low-energy constants. The lattice
QCD data, for various pion masses below $400$ MeV, can be well described up to
a momentum transfer of $\sim 0.6$ GeV. The $\Delta(1232)$ loops contribute
significantly to this agreement. Furthermore, we extract the axial charge and
radius based on the fitted values of the low energy constants. The results are:
$g_A=1.237(74)$ and $\langle r_A^2\rangle =0.263(38)~{\rm fm}^2$. The obtained
coupling $g_A$ is consistent with the experimental value if the uncertainty is
taken into account. The axial radius is below but in agreement with the recent
extraction from neutrino quasi-elastic scattering data on deuterium, which has
large error bars. Up to our current working accuracy, $r_A$ is predicted only
at leading order, i.e., one-loop level. A more precise determination might need
terms of $\mathcal{O}(p^5)$.
