Observations of the redshifted 21-cm line of atomic hydrogen have provided several upper limits on the 21-cm power spectrum and a tentative detection of the sky-averaged signal at redshift z ≈ 17. Made with the Experiment to Detect the Global EoR Signature (EDGES) low-band antenna, this claim was recently disputed by the SARAS 3 experiment, which reported a non-detection and is the only available upper limit strong enough to constrain cosmic dawn astrophysics. We use these data to constrain a population of radio-luminous galaxies ~200 million years after the Big Bang (z ≈ 20). We find, using Bayesian data analysis, that the data disfavour (at 68% confidence) radio-luminous galaxies in dark-matter haloes with masses of 4.4 × 105 M⊙ ≲ M ≲ 1.1 × 107 M⊙ (where M⊙ is the mass of the Sun) at z = 20 and galaxies in which >5% of the gas is converted into stars. The data disfavour galaxies with a radio luminosity per star formation rate of Lr/SFR ≳ 1.549 × 1025 W Hz−1 M⊙−1 yr at 150 MHz, around 1,000 times brighter than today, and, separately, a synchrotron radio background in excess of the cosmic microwave background by ≳6% at 1.42 GHz.