First-quantized deep neural network techniques are developed for analyzing strongly coupled fermionic systems on the lattice. Using a Slater-Jastrow-inspired ansatz which exploits deep residual networks with convolutional residual blocks, we approximately determine the ground state of spinless fermions on a square lattice with nearest-neighbor interactions. The flexibility of the neural-network ansatz results in a high level of accuracy when compared with exact diagonalization results on small systems, both for energy and correlation functions. On large systems, we obtain accurate estimates of the boundaries between metallic and charge-ordered phases as a function of the interaction strength and the particle density.