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Polarimetric observations of the repeating fast radio burst source FRB 121102 suggest that the bursts may originate from a neutron star in the extreme magneto-ionic environment of an accreting massive black hole or supernova remnant.Fast radio bursts are millisecond-duration, extragalactic radio flashes of unknown physical origin1,2,3. The only known repeating fast radio burst source4,5,6—FRB 121102—has been localized to a star-forming region in a dwarf galaxy7,8,9 at redshift 0.193 and is spatially coincident with a compact, persistent radio source7,10. The origin of the bursts, the nature of the persistent source and the properties of the local environment are still unclear. Here we report observations of FRB 121102 that show almost 100 per cent linearly polarized emission at a very high and variable Faraday rotation measure in the source frame (varying from +1.46 × 105 radians per square metre to +1.33 × 105 radians per square metre at epochs separated by seven months) and narrow (below 30 microseconds) temporal structure. The large and variable rotation measure demonstrates that FRB 121102 is in an extreme and dynamic magneto-ionic environment, and the short durations of the bursts suggest a neutron star origin. Such large rotation measures have hitherto been observed11,12 only in the vicinities of massive black holes (larger than about 10,000 solar masses). Indeed, the properties of the persistent radio source are compatible with those of a low-luminosity, accreting massive black hole10. The bursts may therefore come from a neutron star in such an environment or could be explained by other models, such as a highly magnetized wind nebula13 or supernova remnant14 surrounding a young neutron star.