The Search for Hidden Particles (SHiP) Collaboration has proposed a general-purpose experimental facility operating in beam-dump mode at the CERN SPS accelerator to search for light, feebly interacting particles. In the baseline configuration, the SHiP experiment incorporates two complementary detectors. The upstream detector is designed for recoil signatures of light dark matter (LDM) scattering and for neutrino physics, in particular with tau neutrinos. It consists of a spectrometer magnet housing a layered detector system with high-density LDM/neutrino target plates, emulsion-film technology and electronic high-precision tracking. The total detector target mass amounts to about eight tonnes. The downstream detector system aims at measuring visible decays of feebly interacting particles to both fully reconstructed final states and to partially reconstructed final states with neutrinos, in a nearly background-free environment. The detector consists of a 50$\mathrm { \,m}$ long decay volume under vacuum followed by a spectrometer and particle identification system with a rectangular acceptance of 5 m in width and 10 m in height. Using the high-intensity beam of 400$\,\mathrm {GeV}$ protons, the experiment aims at profiting from the $4\times 10^{19}$ protons per year that are currently unexploited at the SPS, over a period of 5–10 years. This allows probing dark photons, dark scalars and pseudo-scalars, and heavy neutral leptons with GeV-scale masses in the direct searches at sensitivities that largely exceed those of existing and projected experiments. The sensitivity to light dark matter through scattering reaches well below the dark matter relic density limits in the range from a few ${\mathrm {\,MeV\!/}c^2}$ up to 100 MeV-scale masses, and it will be possible to study tau neutrino interactions with unprecedented statistics. This paper describes the SHiP experiment baseline setup and the detector systems, together with performance results from prototypes in test beams, as it was prepared for the 2020 Update of the European Strategy for Particle Physics. The expected detector performance from simulation is summarised at the end.