  We study the functional renormalization group of a three-dimensional
tensorial Group Field Theory (GFT) with gauge group SU(2). This model generates
(generalized) lattice gauge theory amplitudes, and is known to be
perturbatively renormalizable up to order 6 melonic interactions. We consider a
series of truncations of the exact Wetterich--Morris equation, which retain
increasingly many perturbatively irrelevant melonic interactions. This
tensorial analogue of the ordinary local potential approximation allows to
investigate the existence of non-perturbative fixed points of the
renormalization group flow. Our main finding is a candidate ultraviolet fixed
point, whose qualitative features are reproduced in all the truncations we have
checked (with up to order 12 interactions). This may be taken as evidence for
an ultraviolet completion of this GFT in the sense of asymptotic safety.
Moreover, this fixed point has a single relevant direction, which suggests the
presence of two distinct infrared phases. Our results generally support the
existence of GFT phases of the condensate type, which have recently been
conjectured and applied to quantum cosmology and black holes.
