Short X-ray reverberation lags are seen across a broad Fe-K energy band in more than twenty active galactic nuclei (AGNs). This broad iron line feature in the lag spectrum is most significant in super-Eddington sources such as Ark 564 ($L/L_{rm Edd}sim 1$) and 1H 0707--495 ($L/L_{rm Edd}gtrsim 10$). The observed lag timescales correspond to very short distances of several $R_g/c$, so that they have been used to argue for extremely small `lamp-post coronae close to the event horizon of rapidly spinning black holes. Here we show for the first time that these Fe-K short lags are more likely to arise from scattering in a highly-ionised wind, launched at $sim 50,R_g$, rotating and outflowing with a typical velocity of $0.2c$. We show that this model can simultaneously fit the time-averaged energy spectra and the short-timescale lag-energy spectra of both 1H 0707--495 and Ark 564. The Fe-K line in 1H 0707--495 has a strong P-Cygni-like profile, which requires that the wind solid angle is large and that our line of sight intercepts the wind. By contrast the lack of an absorption line in the energy spectrum of Ark 564 requires rather face-on geometry, while the weaker broad Fe-K emission in the energy and lag-energy spectra argue for a smaller solid angle of the wind. This is consistent with theoretical predictions that the winds get stronger when the sources are more super-Eddington, supporting the idea of AGN feedback via radiation pressure driven winds.
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