  Disorder-free localization has been recently introduced as a mechanism for
ergodicity breaking in low-dimensional homogeneous lattice gauge theories
caused by local constraints imposed by gauge invariance. We show that also
genuinely interacting systems in two spatial dimensions can become nonergodic
as a consequence of this mechanism. Specifically, we prove nonergodic behavior
in the quantum link model by obtaining a rigorous bound on the
localization-delocalization transition through a classical correlated
percolation problem implying a fragmentation of Hilbert space on the nonergodic
side of the transition. We study the quantum dynamics in this system by means
of an efficient and perturbatively controlled representation of the
wavefunction in terms of a variational network of classical spins akin to
artificial neural networks. We identify a distinguishing dynamical signature by
studying the propagation of line defects, yielding different light cone
structures in the localized and ergodic phases, respectively. The methods we
introduce in this work can be applied to any lattice gauge theory with
finite-dimensional local Hilbert spaces irrespective of spatial dimensionality.
