  In the context of transient constant-roll inflation near a local maximum, we
derive the non-perturbative field redefinition that relates a Gaussian random
field with the true non-Gaussian curvature perturbation. Our analysis shows the
emergence of a new critical amplitude $\zeta_*$, corresponding to perturbations
that prevent the inflaton from overshooting the local maximum, thus becoming
trapped in the false minimum of the potential. For potentials with a mild
curvature at the local maximum (and thus small non-Gaussianity), we recover the
known perturbative field redefinition. We apply these results to the formation
of primordial black holes, and discuss the cases for which $\zeta_*$ is smaller
or of the same order than the critical value for collapse of spherically
symmetric overdensities. In the latter case, we present a simple potential for
which the power spectrum needs an amplitude 10 times smaller that in the
Gaussian case for producing a sizeable amount of primordial black holes.
