This paper presents a novel nonlinear adaptive neural control methodology for spacecraft formation flying in the spatial elliptic restricted three-body problem (ER3BP). By utilizing the framework of the ER3BP with the Sun and Earth as the primaries, a nonlinear non-autonomous model is first developed, which describes the relative formation dynamics. Then, a relative position controller is designed, which consists of an approximate dynamic model inversion, linear compensation of the ideal feedback linearized model, and an adaptive neural network based element designed to compensate for the model inversion errors. The nominal dynamic inversion includes the gravitational forces and the linear non-autonomous terms, while the model inversion errors are assumed to stem from disturbances such as fourth-body gravitational effects and solar radiation pressure. Simulation illustrates the impressive controller performance.