Context. The chromosphere above sunspot umbrae and penumbrae shows several different types of fast dynamic events such as running penumbral waves, umbral flashes, and penumbral microjets. Aims. The aim of this paper is to identify the physical driver responsible for the dynamic and small-scale chromospheric jets above a sunspot light bridge. Methods. High-resolution broadband filtergrams of active region NOAA 11271 in Ca ii H and G band were obtained with the Solar Optical Telescope on board Hinode. We identified the jets in the Ca ii H images using a semi-automatic routine and determined their length and orientation. We applied local correlation tracking (LCT) to the G-band images to obtain the photospheric horizontal velocity field. The magnetic field topology was derived from a Milne-Eddington inversion of a simultaneous scan with the Spectropolarimeter. Results. The chromospheric jets consist of a bright, triangular-shaped blob that lies on the light bridge, while the apex of this blob extends into a spike-like structure that is bright against the dark umbral background. Most of the jets have apparent lengths of less than 1000 km and about 30% of the jets have lengths between 1000–1600 km. The jets are oriented within ±35° to the normal of the spine of the light bridge. Most of them are clustered near the central part of the light bridge within a 2′′ area. The jets are seen to move rapidly along the light bridge and many of them cannot be identified in successive images taken with a 2 min cadence. The jets are primarily located on one side of the light bridge and are directed into the umbral core. The Stokes profiles at or close to the location of the blobs on the LB exhibit both a significant net circular polarization and multiple components, including opposite-polarity lobes. The magnetic field diverges from the light bridge towards the umbral cores that it separates. The LCT reveals that in the photosphere there is a predominantly uni-directional flow with speeds of 100–150 m s-1 along the light bridge. This unidirectional flow is interrupted by a patch of weak or very small motions on the light bridge which also moves along the light bridge. Conclusions. The dynamic short-lived chromospheric jets above the LB seem to be guided by the magnetic field lines. Reconnection events are a likely trigger for such phenomenon since they occur at locations where the magnetic field changes orientation sharply and where we also observe isolated patches of opposite-polarity magnetic components. We find no clear relation between the jets and the photospheric flow pattern.