We experimentally demonstrate a new type of spin Hall nano-oscillator based on a vertical nanocontact fabricated on a Pt/[Co/Ni] multilayer. We analyze the spectral characteristics of the nano-oscillator as a function of current, magnetic field, and temperature. At sufficiently large currents, the oscillator exhibits dynamics at a frequency far below the ferromagnetic resonance, which at large fields exhibits a redshift with increasing current. At smaller fields and low temperatures, the frequency becomes nearly current-independent, with a well-defined threshold current I th. These distinct spectral characteristics of the demonstrated nano-oscillator can be explained by the formation of the magnetic droplet-a dissipative magnetic soliton stabilized by the local injection of spin current produced by the spin Hall effect in Pt. The minimum linewidth exhibits a linear temperature dependence, suggesting single-mode dynamics, and enabling coherent magnetization auto-oscillation at room temperature. The demonstrated nano-oscillator geometry provides new opportunities for the development of active nanomagnetic devices, and optimization of their spectral characteristics for applications in microwave technology and spin-wave logic.