ReflectOrbital is planning to launch a $18\times18$\,m mirror in space at a $\sim$600\,km altitude to illuminate a $2.5$\,km radius circular patch on Earth. The pilot satellite, called EARENDIL-1, is a pathfinder for a constellation of $\sim$50,000 larger mirrors, each with $54\times54$\,m in diameter, with the explicit goal to provide space-borne illumination on Earth. We calculate the sky brightness caused by the Rayleigh scattering and aerosol scattering of the incoming beam from one such satellite and by the light reflected back by the illuminated patch, for various atmospheric properties and ground albedos under cloud-free conditions. We demonstrate that the light pollution caused by one of these satellites is very significant, altering the nighttime environment up to distances of $\sim$30\,kms. The results show that for an observer within the beam of a single 54\,m satellite, the mirror will appear as a $-16.7$\,mag point-like source, i.e.~about 4 magnitudes brighter than the full moon. The diffuse sky background will be similar to the dusk sky shortly after sunset; too bright to see even the brightest stars. From a distance of 14\,km, the glow from a single mirror will exceed the luminance of the full moon sky for majority of the sky. From a distance of 34\,km the sky will still appear brighter than the moon-lit sky in the direction of the beam. If 400 such mirrors illuminate the same patch simultaneously, the glow will be obvious from 80\,kms.