Several interstellar environments produce 'anomalous microwave emission' (AME), with brightness peaks at tens-of-gigahertz frequencies. The emission's origins are uncertain -- rapidly spinning nanoparticles could emit electric-dipole radiation, but the polycyclic aromatic hydrocarbons that have been proposed as the carrier are now found not to correlate with Galactic AME signals. The difficulty is in identifying co-spatial sources over long lines of sight. Here we identify AME in three proto-planetary discs. These are the only known systems that host hydrogenated nanodiamonds, in contrast to the very common detection of polycyclic aromatic hydrocarbons. Using spectroscopy, the nanodiamonds are located close to the host stars, at physically well-constrained temperatures. Developing disc models, we reproduce the emission with diamonds 0.75--1.1 nm in radius, holding <= 1-2% of the carbon budget. Ratios of microwave emission to stellar luminosity are approximately constant, allowing nanodiamonds to be ubiquitous but emitting below detection thresholds in many star systems. This result is compatible with the findings with similar-sized diamonds found within Solar System meteorites. As nanodiamond spectral absorption is seen in interstellar sightlines, these particles are also a candidate for generating galaxy-scale AME.