Cosmological simulations predict that the Universe contains a network of intergalactic gas filaments, within which galaxies form and evolve. However, the faintness of any emission from these filaments has limited tests of this prediction. We report the detection of rest-frame ultraviolet Lyman-α radiation from multiple filaments extending more than one megaparsec between galaxies within the SSA22 protocluster at a redshift of 3.1. Intense star formation and supermassive black-hole activity is occurring within the galaxies embedded in these structures, which are the likely sources of the elevated ionizing radiation powering the observed Lyman-α emission. Our observations map the gas in filamentary structures of the type thought to fuel the growth of galaxies and black holes in massive protoclusters.