Dark matter is a well-established theoretical addition to the Standard Model supported by many observations in modern astrophysics and cosmology. In this context, the existence of weakly interacting massive particles represents an appealing solution to the observed thermal relic in the Universe. Indeed, a large experimental campaign is ongoing for the detection of such particles in the sub-GeV mass range. Adopting the benchmark scenario for light dark matter particles produced in the decay of a dark photon, with α$_{D}$ = 0.1 and m$_{A′}$ = 3m$_{χ}$, we study the potential of the SHiP experiment to detect such elusive particles through its Scattering and Neutrino detector (SND). In its 5-years run, corresponding to 2 · 10$^{20}$ protons on target from the CERN SPS, we find that SHiP will improve the current limits in the mass range for the dark matter from about 1 MeV to 300 MeV. In particular, we show that SHiP will probe the thermal target for Majorana candidates in most of this mass window and even reach the Pseudo-Dirac thermal relic.