Abstract <p>Properties of nonlinear muon-acoustic waves (MAWs) propagating in three-component <i>e</i>^‒μ^‒<i>i</i>^<i>+</i>‑plasma were studied. Linear analysis of MAWs was performed and the dispersion equation was derived. The analysis performed resulted in obtaining the exact expression for the linear velocity of muon-sound waves. By means of normalizing soliton velocity to this expression, its Mach number was obtained. Nonlinear analysis was performed involving the Sagdeev pseudopotential method. It was found that in the <i>e</i>^‒μ^‒<i>i</i>⁺-plasma, the stationary periodic subsonic MAWs can propagate, as well as supersonic muon-acoustic solitons (MA-solitons). It was also found that in MAWs, the muon and electron plasma components change in antiphase with each other. It was shown that in the <i>e</i>^‒μ^‒<i>i</i>⁺-plasma, in the case of relative electron density being equal to unity or somewhat less, the Mach numbers of muon-acoustic solitons can be abnormally high. Profiles of physical parameters in such hypersonic MA-solitons were calculated. It was demonstrated that the amplitudes of physical parameters characterizing hypersonic MA-solitons can be giant.</p>