Context. Resistive Ohmic dissipation has been suggested as a mechanism for heating the solar chromosphere, but few studies have established this association. Aims. We aim to determine how Ohmic dissipation by electric currents can heat the solar chromosphere. Methods. We combine high-resolution spectroscopic Ca II data from the Dunn Solar Telescope and vector magnetic field observations from the Helioseismic and Magnetic Imager (HMI) to investigate thermal enhancements in a sunspot light bridge. The photospheric magnetic field from HMI was extrapolated to the corona using a non-force-free field technique that provided the three-dimensional distribution of electric currents, while an inversion of the chromospheric Ca II line with a local thermodynamic equilibrium and a nonlocal thermodynamic equilibrium spectral archive delivered the temperature stratifications from the photosphere to the chromosphere. Results. We find that the light bridge is a site of strong electric currents, of about 0.3 A m⁻² at the bottom boundary, which extend to about 0.7 Mm while decreasing monotonically with height. These currents produce a chromospheric temperature excess of about 600−800 K relative to the umbra. Only the light bridge, where relatively weak and highly inclined magnetic fields emerge over a duration of 13 h, shows a spatial coincidence of thermal enhancements and electric currents. The temperature enhancements and the Cowling heating are primarily confined to a height range of 0.4−0.7 Mm above the light bridge. The corresponding increase in internal energy of 200 J m⁻³ can be supplied by the heating in about 10 min. Conclusions. Our results provide direct evidence for currents heating the lower solar chromosphere through Ohmic dissipation.