  The gravitational lensing signal from the Cosmic Microwave Background is
highly valuable to constrain the growth of the structures in the Universe in a
clean and robust manner over a wide range of redshifts. One of the theoretical
systematics for lensing reconstruction is the impact of the lensing field
non-Gaussianities on its estimators. Non-linear matter clustering and post-Born
lensing corrections are known to bias standard quadratic estimators to some
extent, most significantly so in temperature. In this work, we explore the
impact of non-Gaussian deflections on Maximum a Posteriori lensing estimators,
which, in contrast to quadratic estimators, are able to provide optimal
measurements of the lensing field. We show that these naturally reduce the
induced non- Gaussian bias and lead to unbiased cosmological constraints in
$\Lambda$CDM at CMB-S4 noise levels without the need for explicit modelling. We
also test the impact of assuming a non-Gaussian prior for the reconstruction;
this mitigates the effect further slightly, but generally has little impact on
the quality of the reconstruction. This shows that higher-order statistics of
the lensing deflections are not expected to present a major challenge for
optimal CMB lensing reconstruction in the foreseeable future.
