We propose a new mission called Space Project for Astrophysical and Cosmological Exploration (SPACE) as part of the ESA long term planning Voyage 2050 programme. SPACE will study galaxy evolution at the earliest times, with the key goals of charting the formation of the heavy elements, measuring the evolution of the galaxy luminosity function, tracing the build-up of stellar mass in galaxies over cosmic time, and finding the first super-massive black holes (SMBHs) to form. The mission will exploit a unique region of the parameter space, between the narrow ultra-deep surveys with HST and JWST, and shallow wide-field surveys such as the Roman Space Telescope and EUCLID, and should yield by far the largest sample of any current or planned mission of very high redshift galaxies at z > 10 which are sufficiently bright for detailed follow-up spectroscopy. Crucially, we propose a wide-field spectroscopic near-IR + mid-IR capability which will greatly enhance our understanding of the first galaxies by detecting and identifying a statistical sample of the first galaxies and the first supermassive black holes, and to chart the metal enrichment history of galaxies in the early Universe - potentially finding signatures of the very first stars to form from metal-free primordial gas. The wide-field and wavelength range of SPACE will also provide us a unique opportunity to study star formation by performing a wide survey of the Milky Way in the near-IR + mid-IR. This science project can be enabled either by a stand-alone ESA-led M mission or by an instrument for an L mission (with ESA and/or NASA, JAXA and other international space agencies) with a wide-field (sub-)millimetre capability at λ > 500 μm.