Levitated particles are unique among optomechanical systems in that they benefit from the absence of physical contact with the external environment. Recently, a new research direction known as levitated optomechanics has attracted interest in numerous research groups, with a major focus on optically suspended particles. In contrast to optical trapping experiments, we levitate charged silica nanospheres in high vacuum by means of a Paul trap. This method provides a deeper confining potential than that of optical traps and enables trapping of optically opaque objects. A detection system based on back-focal-plane interferometry allows us to observe center-of-mass (CoM) motion of the particle. We introduce an additional laser beam that is focused on the particle and provides optical forces with projections on all three principal axes of the Paul trap. This additional beam is intensity-modulated by an acousto-optic modulator controlled by feedback electronics. In this way, we are able to cool the secular motion of the CoM below 1 K, the effective temperature in all three directions being currently limited only by the detection efficiency. This is the first time, to the best of our knowledge, that laser cooling of mechanical motion of a nanoparticle in a Paul trap potential has been demonstrated. Such cooling acts locally on a single particle, in contrast to feedback provided by auxiliary electric fields, and opens up possibilities for sympathetic cooling of particles levitated in Paul traps when other methods are not suitable, for example, in the case of highly absorptive particles.