The interaction of matter–wave solitons with a potential barrier is a fundamentally important problem, and the splitting and subsequent recombination of the soliton by the barrier is the essence of soliton matter–wave interferometry. We demonstrate the three-dimensional (3D) character of the interactions in the case relevant to ongoing experiments, where the number of atoms in the soliton is relatively close to the collapse threshold. We examine the soliton dynamics in the framework of the effectively one-dimensional (1D) nonpolynomial Schr¨odinger equation (NPSE), which admits the collapse in a modified form, and in parallel we use the full 3D Gross–Pitaevskii equation (GPE). Both approaches produce similar results, which are, however, quite different from those produced in recent work that used the 1D cubic GPE. Basic features, produced by the NPSE and the 3D GPE alike, include (a) an increase in the first reflection coefficient for increasing barrier height and decreasing atom number; (b) large variation of the secondary reflection/recombination probability versus barrier height; (c) pronounced asymmetry in the oscillation amplitudes of the transmitted and reflected fragments; and (d) enhancement of the transverse excitations as the number of atoms is increased. We also explore effects produced by variations of the barrier width and outcomes of the secondary collision upon phase imprinting on the fragment in one arm of the interferometer.