  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.
