We report a centimetre-wave (cm-wave, 5–31 GHz) excess over free–free emission in planetary nebulae (PNe). Accurate 31- and 250-GHz measurements show that the 31-GHz flux densities in our sample are systematically higher than the level of optically thin free–free continuum extrapolated from 250-GHz. The 31-GHz excess is observed, within one standard deviation, in all 18 PNe with reliable 31- and 250-GHz data, and is significant in nine PNe. The only exception is the peculiar object M2-9, whose radio spectrum is that of an optically thick stellar wind. On average the fraction of non-free–free emission represents 51 per cent of the total flux density at 31 GHz, with a scatter of 11 per cent. The average 31–250 GHz spectral index of our sample is 〈α25031〉=−0.43 ± 0.03 (in flux density, with a scatter of 0.14). The 31–250 GHz drop is reminiscent of the anomalous foreground observed in the diffuse interstellar medium (ISM) by cosmic microwave background (CMB) anisotropy experiments. The 5–31 GHz spectral indices are consistent with both flat spectra and spinning dust emissivities, given the 10 per cent calibration uncertainty of the comparison 5-GHz data. However, a detailed study of the objects with the largest cm-excess, including the low-frequency data available in the literature, shows that present spinning dust models alone cannot explain the cm-wave excess in PNe. Although we have no definitive interpretation of our data, the least implausible explanation involves a synchrotron component absorbed by a cold nebular screen. We give flux densities for 37 objects at 31 GHz, and for 26 objects at 250 GHz.