Several interstellar environments produce anomalous microwave emission, with brightness-peaks at tens-of-gigahertz frequencies. The emissions origins are uncertain - rapidly-spinning nano-particles could emit electric-dipole radiation, but polycyclic aromatic hydrocarbons proposed as the carrier are now found not to correlate with Galactic signals. The difficulty is to identify co-spatial sources over long lines of sight. Here we identify anomalous microwave emission in three proto-planetary discs. These are the only known systems that host hydrogenated nano-diamonds, in contrast to very common detection of polycyclic aromatic hydrocarbons. Spectroscopy locates the nano-diamonds close to the host-stars, at physically-constrained temperatures. Developing disc models, we reproduce the emission with diamonds 0.75-1.1 nanometres in radius, holding less than or equal to 1-2 per cent of the carbon budget. The microwave-emission:stellar-luminosity ratios are approximately constant, allowing nano-diamonds to be ubiquitous but emitting below detection thresholds in many star-systems. This can unify the findings with similar-sized diamonds found within solar system meteorites. As nano-diamond spectral absorption is seen in interstellar sightlines, these particles are also a candidate for generating galaxy-scale anomalous microwave emission.
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