ABSTRACT We use the XMM–Newton EPIC‐pn instrument in timing mode to extend spectral time‐lag studies of hard state black hole X‐ray binaries into the soft X‐ray band. We show that variations of the disc blackbody emission substantially lead variations in the power‐law emission, by tenths of a second on variability time‐scales of seconds or longer. The large lags cannot be explained by Compton scattering but are consistent with time delays due to viscous propagation of mass accretion fluctuations in the disc. However, on time‐scales less than a second the disc lags the power‐law variations by a few milliseconds, consistent with the disc variations being dominated by X‐ray heating by the power law, with the short lag corresponding to the light traveltime between the power‐law emitting region and the disc. Our results indicate that instabilities in the accretion disc are responsible for continuum variability on time‐scales of seconds or longer and probably also on shorter time‐scales.