Short X-ray reverberation lags are seen across a broad Fe-K energy band in more than 20 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-Edd similar to 1) and 1H 0707-495 (L/L-Edd greater than or similar to 10). The observed lag time-scales 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 ionized wind, launched at similar to 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-time-scale 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.