We present a detailed analysis of the kinematic properties of the multiple populations (MPs) in the low-mass Galactic globular cluster NGC 6362 based on a sample of about 500 member stars for which radial velocities (RVs), Fe and Na abundances have been homogeneously derived. At distances from the cluster center larger than about 0.5r h , we find that first (FG-Na-poor) and second generation (SG-Na-rich) stars show hints of different line-of-sight velocity dispersion profiles, with FG stars being dynamically hotter. This is the first time that differences in the velocity dispersion of MPs are detected by using only RVs. While kinematic differences between MPs in globular clusters are usually described in terms of anisotropy differences driven by the different radial distributions, this explanation seems hardly viable for NGC 6362, where SG and FG stars are spatially mixed. We demonstrate that the observed difference in the velocity dispersion profiles can be accounted for by the effect of binary stars. In fact, thanks to our multi-epoch RV measurements we find that the binary fraction is significantly larger in the FG sample (f ∼ 14%) than in SG population (f < 1%), and we show that such a difference can inflate the velocity dispersion of FG with respect to SG by the observed amount in the relevant radial range. Our results nicely match the predictions of state-of-the art N-body simulations of the co-evolution of MPs in globular clusters that include the effect of binaries.