Fast radio bursts (FRBs) are millisecond-duration, extragalactic radio flashes of unknown physical origin. FRB 121102, the only known repeating FRB source, has been localized to a star-forming region in a dwarf galaxy at redshift z = 0.193, and is spatially coincident with a compact, persistent radio source. The origin of the bursts, the nature of the persistent source, and the properties of the local environment are still debated. Here we present bursts that show ~100% linearly polarized emission at a very high and variable Faraday rotation measure in the source frame: RM_src = +1.46 x 10^5 rad m^-2 and +1.33 x 10^5 rad m^-2 at epochs separated by 7 months, in addition to narrow (< 30 mus) temporal structure. The large and variable rotation measure demonstrates that FRB 121102 is in an extreme and dynamic magneto-ionic environment, while the short burst durations argue for a neutron star origin. Such large rotation measures have, until now, only been observed in the vicinities of massive black holes (M_BH > 10^4 MSun). Indeed, the properties of the persistent radio source are compatible with those of a low-luminosity, accreting massive black hole. The bursts may thus come from a neutron star in such an environment. However, the observed properties may also be explainable in other models, such as a highly magnetized wind nebula or supernova remnant surrounding a young neutron star.
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