We present in this paper an extensive study of the size dependence of the low-frequency vibrational eigenmodes of two- and thr ee-dimensional amorphous (non-crystalline) nanoparticles. We show that classical continuum elasticity (Lamb modes) breaks down when the wavelength of the solicitation is smaller than a characteristic length of approximately 40 atomic sizes. This length is related to the presence of additional vortices in the atomic displacement field of the nanoparticle. These vortices are re lated to local anisotropy in the inhomogeneous elastic moduli of the amorphous structure. We show how they affect the average mechanical response of the nanoparticles.