Radio pulsars scintillate because their emission travels through the ionized interstellar medium along multiple paths, which interfere with each other. It has long been realized that, independent of their nature, the regions responsible for the scintillation could be used as 'interstellar lenses' to localize pulsar emission regions1,2. Most such lenses, however, resolve emission components only marginally, limiting results to statistical inferences and detections of small positional shifts3-5. As lenses situated close to their source offer better resolution, it should be easier to resolve emission regions of pulsars located in high-density environments such as supernova remnants 6 or binaries in which the pulsar's companion has an ionized outflow. Here we report observations of extreme plasma lensing in the 'black widow' pulsar, B1957+20, near the phase in its 9.2-hour orbit at which its emission is eclipsed by its companion's outflow7-9. During the lensing events, the observed radio flux is enhanced by factors of up to 70-80 at specific frequencies. The strongest events clearly resolve the emission regions: they affect the narrow main pulse and parts of the wider interpulse differently. We show that the events arise naturally from density fluctuations in the outer regions of the outflow, and we infer a resolution of our lenses that is comparable to the pulsar's radius, about 10 kilometres. Furthermore, the distinct frequency structures imparted by the lensing are reminiscent of what is observed for the repeating fast radio burst FRB 121102, providing observational support for the idea that this source is observed through, and thus at times strongly magnified by, plasma lenses 10 .