Optical trapping at high vacuum of a nanodiamond containing a nitrogen vacancy centre would provide a test bed for several new phenomena in fundamental physics. However, the nanodiamonds used so far have absorbed too much of the trapping light, heating them to destruction (above 800 K) except at pressures above ∼10 mbar where air molecules dissipate the excess heat. Here we show that milling diamond of 1000 times greater purity creates nanodiamonds that do not heat up even when the optical intensity is raised above 700 GW m −2 below 5 mbar of pressure.