X-rays illuminating the accretion disc in active galactic nuclei give rise to
an iron K line and its associated reflection spectrum which are lagged behind
the continuum variability by the light-travel time from the source to the disc.
The measured lag timescales in the iron band can be as short as $\sim R_g/c$,
where $R_g$ is the gravitational radius, which is often interpreted as evidence
for a very small continuum source close to the event horizon of a rapidly
spinning black hole. However, the short lags can also be produced by reflection
from more distant material, because the primary photons with no time-delay
dilute the time-lags caused by the reprocessed photons. We perform a
Monte-Carlo simulation to calculate the dilution effect in the X-ray
reverberation lags from a half-shell of neutral material placed at $100\,R_g$
from the central source. This gives lags of $\sim2\,R_g/c$, but the iron line
is a distinctly narrow feature in the lag-energy plot, whereas the data often
show a broader line. We show that both the short lag and the line broadening
can be reproduced if the scattering material is outflowing at $\sim0.1c$. The
velocity structure in the wind can also give shifts in the line profile in the
lag-energy plot calculated at different frequencies. Hence we propose that the
observed broad iron reverberation lags and shifts in profile as a function of
frequency of variability can arise from a disc wind at fairly large distances
from the X-ray source.