  We discuss the potential of a next generation space-borne Cosmic Microwave
Background (CMB) experiment for studies of extragalactic sources. Our analysis
has particular bearing on the definition of the future space project, CORE,
that has been submitted in response to ESA's call for a Medium-size mission
opportunity as the successor of the Planck satellite. Even though the effective
telescope size will be somewhat smaller than that of Planck, CORE will have a
considerably better angular resolution at its highest frequencies, since, in
contrast with Planck, it will be diffraction limited at all frequencies. The
improved resolution implies a considerable decrease of the source confusion,
i.e. substantially fainter detection limits. In particular, CORE will detect
thousands of strongly lensed high-z galaxies distributed over the full sky. The
extreme brightness of these galaxies will make it possible to study them, via
follow-up observations, in extraordinary detail. Also, the CORE resolution
matches the typical sizes of high-z galaxy proto-clusters much better than the
Planck resolution, resulting in a much higher detection efficiency; these
objects will be caught in an evolutionary phase beyond the reach of surveys in
other wavebands. Furthermore, CORE will provide unique information on the
evolution of the star formation in virialized groups and clusters of galaxies
up to the highest possible redshifts. Finally, thanks to its very high
sensitivity, CORE will detect the polarized emission of thousands of radio
sources and, for the first time, of dusty galaxies, at mm and sub-mm
wavelengths, respectively.
