We introduce a message-passing neural-network (NN)-based wave function Ansatz to simulate extended, strongly interacting fermions in continuous space. Symmetry constraints, such as continuous translation symmetries, can be readily embedded in the model. We demonstrate its accuracy by simulating the ground state of the homogeneous electron gas in three spatial dimensions at different densities and system sizes. With orders of magnitude fewer parameters than state-of-the-art NN wave functions, we demonstrate better or comparable ground-state energies. Reducing the parameter complexity allows scaling to N=128 electrons, previously inaccessible to NN wave functions in continuous space, allowing future work on finite-size extrapolations to the thermodynamic limit. We also show the capability of the Ansatz to quantitatively represent different phases of matter.