Context. The plasma near the nucleus of a comet is subjected to an electric field to which a few different sources contribute: the convective electric field of the solar wind, the ambipolar electric field due to higher electron than ion speeds, and a polarisation field arising from the vastly different ion and electron trajectories. Aims. Our aim is to show how the ambipolar and polarisation electric fields arise and develop under the influence of space charge effects, and in doing so we paint a qualitative picture of the electric fields in the inner coma of a comet. Methods. We use an electrostatic particle-in-cell model to simulate a scaled-down comet, representing comet 67P/Churyumov-Gerasimenko with parameters corresponding to a 3.0 AU heliocentric distance. Results. We find that an ambipolar electric field develops early in the simulation and that this is soon followed by the emergence of a polarisation electric field, manifesting itself as an anti-sunward component prevalent in the region surrounding the centre of the comet. As plasma is removed from the inner coma in the direction of the convectional electric field of the solar wind, a density maximum develops on the opposite side of the centre of the comet. Conclusions. The ambipolar and polarisation electric fields both have a significant influence on the motion of cometary ions. This demonstrates the importance of space charge effects in comet plasma physics.