Fully differential cross sections for single ionization of helium by fast proton impact in different kinematical regimes in the scattering plane were recently measured in a high-precision experiment [O. Chuluunbaatar et al., Phys. Rev. A 99, 062711 (2019)] and calculated using the first Born approximation. We use the nonperturbative wave-packet convergent close-coupling approach to calculate this process more accurately in all the kinematical regimes considered in the experiment. The obtained results show that the coupling between channels and multiple-scattering effects, combined with a more accurate treatment of the target structure, significantly improves the agreement between theory and experiment, especially in the apparently most difficult regions away from the so-called Bethe ridge, where the deviation in the positions of the binary peak observed in the experiment and calculated using the first Born approximation is largest. We also present fully differential cross sections in the same kinematical regimes but for incident projectile energies of 500 keV and 2 MeV. Corresponding results for the so-called perpendicular and azimuthal planes are also exhibited.