Tidal forces close to massive black holes can violently disrupt stars that make a close approach. These extreme events are discovered via bright X-ray and optical/ultraviolet flares in galactic centres. Prior studies based on modelling decaying flux trends have been able to estimate broad properties, such as the mass accretion rate. Here we report the detection of flows of hot, ionized gas in high-resolution X-ray spectra of a nearby tidal disruption event, ASASSN-14li in the galaxy PGC 043234. Variability within the absorption-dominated spectra indicates that the gas is relatively close to the black hole. Narrow linewidths indicate that the gas does not stretch over a large range of radii, giving a low volume filling factor. Modest outflow speeds of a few hundred kilometres per second are observed; these are below the escape speed from the radius set by variability. The gas flow is consistent with a rotating wind from the inner, super-Eddington region of a nascent accretion disk, or with a filament of disrupted stellar gas near to the apocentre of an elliptical orbit. Flows of this sort are predicted by fundamental analytical theory and more recent numerical simulations.