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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., The intrinsic nature and dynamics of a Bloch point has not been verified so far. Here, Im et al. report the realization and dynamical character of steady-state Bloch points in the magnetic vortex cores in asymmetrically shaped Ni80Fe20 nanodisks.