Magnetic thin films with perpendicular magnetic anisotropy have localized excitations that correspond to reversed, dynamically precessing magnetic moments, which are known as magnetic droplet solitons. Fundamentally, these excitations are associated with an attractive interaction between elementary spin-excitations and have been predicted to occur in perpendicularly magnetized materials in the absence of damping. Although damping suppresses these excitations, it can be compensated by spin-transfer torques when an electrical current flows in nanocontacts to ferromagnetic thin films. Theory predicts the appearance of magnetic droplet solitons in nanocontacts at a threshold current and, recently, experimental signatures of droplet nucleation have been reported. However, to date, these solitons have been observed to be nearly reversible excitations, with only partially reversed magnetization. Here, we show that magnetic droplet solitons exhibit a strong hysteretic response in field and current, proving the existence of bistable states: droplet and non-droplet states. In the droplet soliton state we find that the magnetization in the contact is almost fully reversed. These observations, in addition to their fundamental interest, are important to understanding and controlling droplet motion, nucleation and annihilation.