A Bloch point (BP) is a topological defect in a ferromagnet at which the local magnetization vanishes. With the difficulty of generating a stable BP in magnetic nanostructures, the intrinsic nature of a BP and its dynamic behaviour has not been verified experimentally. We report a realization of steady-state BPs embedded in deformed magnetic vortex cores in asymmetrically shaped Ni₈₀Fe₂₀ nanodisks. Time-resolved nanoscale magnetic X-ray imaging combined with micromagnetic simulation shows detailed dynamic character of BPs, revealing rigid and limited lateral movements under magnetic field pulses as well as its crucial role in vortex-core dynamics. Direct visualizations of magnetic structures disclose the unique dynamical feature of a BP as an atomic scale discrete spin texture and allude its influence on the neighbouring spin structures such as magnetic vortices.