The controlled manipulation of spins on ultrashort timescales is among the most promising solutions for novel high-speed and low-power-consumption spintronic and magnetic recording applications. To do so, terahertz (THz) light pulses can be used to drive coherent magnetization dynamics in ferromagnetic thin films. We were able to resolve these dynamics on the nanoscale employing THz-pump x-ray resonant magnetic scattering from the labyrinth-type domain network of a Co/Pt multilayer with perpendicular magnetic anisotropy. Our results reveal THz-driven ultrafast demagnetization as well as coherent local magnetization oscillations at the THz fundamental frequency of 2.5THz. We observe a temporal lag between femtosecond demagnetization and the start of the coherent magnetization oscillations that can be understood by a time-dependent damping. The dynamics of the domain and domain-wall contributions are found to be highly correlated, suggesting the applicability of THz spin control in magnetic nanostructures.