Solid solution and long-range atomic order hardening represent two well-known strategies for increasing strength of bulk metallic materials. Here, we observed the opposite trends in mechanical behavior of defect-lean nanoparticles of Cu-Au alloys fabricated by solid state dewetting of Cu-Au bilayers deposited on sapphire substrate. In the fully disordered state, the nanoparticles of Cu3Au alloy exhibited the highest compressive strength reaching staggering 41 GPa for the smallest studied particles, followed by nanoparticles of pure Cu, and of CuAu alloy. Thus, taking pure Cu and Au nanoparticles as a reference, both solid solution hardening (Cu3Au) and solid solution softening (CuAu) were observed in nanoscale plasticity of the Cu-Au system. Additional annealings at the temperature of 350 °C, 40 °C below the critical point of A1-L12 transformation, were performed to induce different degrees of the long-range order in the Cu3Au nanoparticles. It was shown that the strength of fully ordered Cu3Au nanoparticles is very similar to that of their fully disordered counterparts, whereas the partially ordered Cu3Au nanoparticles larger than 500 nm in size exhibited a significant drop in strength. No effect of long-range order on compressive strength of smaller nanoparticles was observed. These findings were discussed in terms of plasticity mechanisms controlled by nucleation of new dislocations. Specifically, it was proposed that the presence of oxidation-induced Au-rich subsurface layer in the Cu3Au nanoparticles is responsible for their ultrahigh strength.