Magnetic nanoparticles such as FePt in the L10 phase are the bedrock of our current data storage technology. As the grains become smaller to keep up with technological demands, the superparamagnetic limit calls for materials with higher magnetocrystalline anisotropy. This, in turn, reduces the magnetic exchange length to just a few nanometers, enabling magnetic structures to be induced within the nanoparticles. Here, we describe the existence of spin-wave solitons, dynamic localized bound states of spin-wave excitations, in FePt nanoparticles. We show with time-resolved x-ray diffraction and micromagnetic modeling that spin-wave solitons of sub-10 nm sizes form out of the demagnetized state following femtosecond laser excitation. The measured soliton spin precession frequency of 0.1 THz positions this system as a platform to develop novel miniature devices.