We present evidence of a BCS superfluid phase in the $3+1$ dimensional Nambu--Jona-Lasinio model at non-zero chemical potential $\mu$. Large-$N_c$ results are used to match the model's parameters to low-energy, zero-density phenomenology. A lattice study of the fermionic sector yields the quasi-particle dispersion relation, which shows clear evidence of a BCS gap $\Delta\neq0$ in the high-$\mu$ phase. We also discuss the finite volume effects of a system with a Fermi surface.