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Sunspot chromospheres display vigorous oscillatory signature when observed in chromospheric diagnostics like the strong Ca II lines and H-alpha. New high-resolution sunspot observations from the Swedish 1-m Solar Telescope show the ubiquitous presence of small-scale periodic jet-like features that move up and down. This phenomenon has not been described before. Their typical width is about 0.3 arcsec and they display clear parabolic trajectories in space-time diagrams. The maximum extension of the top of the jets is lowest in the umbra, a few 100 km, and progressively longer further away from the umbra in the penumbra, with the longest more than 1000 km. These jets resemble dynamic fibrils found in plage regions but at smaller extensions. LTE inversion of spectro-polarimetric Ca II 8542 observations enabled for a comparison of the magnetic field inclination and the properties of these short jets. We find that the most extended of these jets also have longer periods and tend to be located in regions with more horizontal magnetic fields. This is a direct observational confirmation of the mechanism of long-period waves propagating along inclined magnetic fields into the solar chromosphere. This mechanism was identified earlier as the driver of dynamic fibrils in plage, part of the mottles in quiet Sun, and type I spicules at the limb. The sunspot dynamic fibrils that we report here represent a new class of manifestation of this mechanism. They are not the same as the transient penumbral and umbral micro-jets reported earlier.