Lightning and thunderclouds are natural particle accelerators(1). Avalanches of relativistic runaway electrons, which develop in electric fields within thunderclouds(2,3), emit bremsstrahlung gamma-rays. These gamma-rays have been detected by ground-based observatories(4-9), by airborne detectors(10) and as terrestrial gamma-ray flashes from space(10-14). The energy of the gamma-rays is sufficiently high that they can trigger atmospheric photonuclear reactions(10,15-19) that produce neutrons and eventually positrons via beta(+) decay of the unstable radioactive isotopes, most notably N-13, which is generated via N-14 + gamma -> N-13 + n, where gamma denotes a photon and n a neutron. However, this reaction has hitherto not been observed conclusively, despite increasing observational evidence of neutrons(7,20,21) and positrons(10,22) that are presumably derived from such reactions. Here we report ground-based observations of neutron and positron signals after lightning. During a thunderstorm on 6 February 2017 in Japan, a gamma-ray flash with a duration of less than one millisecond was detected at our monitoring sites 0.5-1.7 kilometres away from the lightning. The subsequent gamma-ray afterglow subsided quickly, with an exponential decay constant of 40-60 milliseconds, and was followed by prolonged line emission at about 0.511 megaelectronvolts, which lasted for a minute. The observed decay time-scale and spectral cutoff at about 10 megaelectronvolts of the gamma-ray afterglow are well explained by de-excitation gamma-rays from nuclei excited by neutron capture. The centre energy of the prolonged line emission corresponds to electron-positron annihilation, providing conclusive evidence of positrons being produced after the lightning.