We use data from the nominal Planck mission to constrain modulations in the primordial power spectrum associated with monodromy inflation. The largest improvement in fit relative to the unmodulated model has \Delta\chi^2~10 and we find no evidence for a primordial signal, in contrast to a previous analysis of the WMAP9 dataset, for which \Delta\chi^2~20. The Planck and WMAP9 results are broadly consistent on angular scales where they are expected to agree as far as best-fit values are concerned. However, even on these scales the significance of the signal is reduced in Planck relative to WMAP, and is consistent with a fit to the noise associated with cosmic variance. Our results motivate both a detailed comparison between the two experiments and a more careful study of the theoretical predictions of monodromy inflation.