Laser cooling has given a boost to atomic physics throughout the last thirty years since it allows one to prepare atoms in motional states which can only be described by quantum mechanics. Most methods, such as Doppler cooling, polarization gradient cooling or sub-recoil laser cooling rely, however, on a near-resonant and cyclic coupling between laser light and well-defined internal states. Although this feat has recently even been achieved for diatomic molecules, it is very hard for mesoscopic particles. It has been proposed that an external cavity may compensate for the lack of internal cycling transitions in dielectric objects and it may thus provide assistance in the cooling of their centre of mass state. Here, we demonstrate cavity cooling of the transverse kinetic energy of silicon nanoparticles propagating in genuine high-vacuum (< 10^8 mbar). We create and launch them with longitudinal velocities even down to v < 1 m/s using laser induced thermomechanical stress on a pristine silicon wafer. The interaction with the light of a high-finesse infrared cavity reduces their transverse kinetic energy by more than a factor of 30. This is an important step towards new tests of recent proposals to explore the still speculative non-linearities of quantum mechanics with objects in the mass range between 10^7 and 10^10 amu.