ABSTRACT The Small and Large Magellanic Clouds are the only galaxies outside our own in which radio pulsars have been discovered to date. The sensitivity of the MeerKAT radio interferometer offers an opportunity to search for a population of more distant extragalactic pulsars. The TRAPUM (TRansients And PUlsars with MeerKAT) collaboration has performed a radio-domain search for pulsars and transients in the dwarf star-forming galaxies Sextans A and B, situated at the edge of the Local Group 1.4 Mpc away. We conducted three 2-h multibeam observations at L band (856–1712 MHz) with the full array of MeerKAT. No pulsars were found down to a radio pseudo-luminosity upper limit of 7.9$\pm$0.4 Jy kpc$^{2}$ at 1400 MHz, which is 28 times more sensitive than the previous limit from the Murriyang telescope. This luminosity is 30 per cent greater than that of the brightest known radio pulsar and sets a cut-off on the luminosity distributions of the entire Sextans A and B galaxies for unobscured radio pulsars beamed in our direction. A fast radio burst was detected in one of the Sextans A observations at a dispersion measure (DM) of 737 pc cm$^{-3}$. We believe this is a background event not associated with the dwarf galaxy due to its large DM and its signal-to-noise ratio being strongest in the wide-field incoherent beam of MeerKAT.