Context. Penumbral filaments and light bridges are prominent structures inside sunspots and are important for understanding the nature of sunspot magnetic fields and magneto-convection underneath. Aims. We investigate an interesting event where several penumbral filaments intrude into a sunspot light bridge. In doing so we aim to gain further insight into the magnetic fields of the sunspot penumbral filament and light bridge, as well as their interaction. Methods. Combining data from the New Vacuum Solar Telescope, Solar Dynamics Observatory, and Interface Region Imaging Spectrograph, we study the emission, kinematic, and magnetic topology characteristics of the penumbral filaments intruding into the light bridge and the resultant jets. Results. At the west part of the light bridge, the intruding penumbral filaments penetrate into the umbrae on both sides of the light bridge, and two groups of jets are also detected. The jets share the same projected morphology with the intruding filaments and are accompanied by intermittent footpoint brightenings. Simultaneous spectral imaging observations provide convincing evidence for the presences of magnetic-reconnection-related heating and bidirectional flows near the jet bases and contribute to measuring the vector velocities of the jets. Additionally, nonlinear force-free field extrapolation results reveal strong and highly inclined magnetic fields along the intruding penumbral filaments, highly consistent with the results deduced from the vector velocities of the jets. Therefore, we propose that the jets could be caused by magnetic reconnections between emerging fields within the light bridge and the nearly horizontal fields of intruding filaments. The jets are then ejected outward along the stronger filament fields. Conclusions. Our study indicates that magnetic reconnection could occur between the penumbral filament fields and emerging fields within the light bridge and produce jets along the stronger filament fields. These results further complement the study of magnetic reconnection and dynamic activities within the sunspot.