Air showers produced by neutral particles, presumably γ-rays, have now been detected from Cygnus X-3 at TeV energies and above 1015 eV. Photons of energy above 1014 eV will suffer pair-production interactions with photons of the microwave background, and the γ-ray flux from sources further than several kpc away will be attenuated. Recent observations of high-energy muons from Cygnus X-3 in underground nucleon decay detectors have led, together with arguments based on the attenuation of γ-rays, to the suggestion that the primary particles responsible for the air showers may not be γ-rays. The recent detection of ultra-high-energy ‘γ-rays’ from a source much further away, in the Large Magellanic Cloud, and an apparent excess of air showers above 1015 eV from the direction of Centaurus A is even more surprising. If, however, the intergalactic magnetic field is sufficiently low, electron–photon cascading in the microwave background may enable distant γ-ray sources such as those in other galaxies to be detected above 1015 eV energies even though the interaction length of γ-rays at 1015 eV is less than 10 kpc. An exact Monte Carlo cascade simulation has been performed to investigate the propagation of γ-rays over large distances to see what effect this will have on the observed spectra of extragalactic sources. The implications of these results for future observations are discussed.