The sensitivity of the MeerKAT radio interferometer is an opportunity to probe deeper into the population of rare and faint extragalactic pulsars. The TRAPUM (TRAnsients and PUlsars with MeerKAT) collaboration has conducted a radio-domain search for accelerated pulsars and transients in the Small Magellanic Cloud (SMC). This partially targeted survey, performed at L-band (856–1712 MHz) with the core array of the MeerKAT telescope in 2-h integrations, is twice as sensitive as the latest SMC radio pulsar survey. We report the discovery of seven new SMC pulsars, doubling this galaxy’s radio pulsar population and increasing the total extragalactic population by nearly a quarter. We also carried out a search for accelerated millisecond pulsars in the SMC Globular Cluster NGC 121 using the full array of MeerKAT. This improved the previous upper limit on pulsed radio emission from this cluster by a factor of six. Our discoveries reveal the first radio pulsar-PWN systems in the SMC, with only one such system previously known outside our galaxy (the ‘Crab pulsar twin’ in the Large Magellanic Cloud, PSR J0540−6919). We associate the 59 ms pulsar discovery PSR J0040−7337, now the fastest spinning radio pulsar in the SMC, with the bow-shock Pulsar Wind Nebula (PWN) of Supernova Remnant DEM S5. We also present a new young pulsar with a 79 ms period, PSR J0048−7317, in a PWN recently discovered in a MeerKAT radio continuum image. Using the multi-beam capability of MeerKAT, we localised our pulsar discoveries, and two previous Murriyang discoveries, to a positional uncertainty of a few arcseconds.