Chemical forces on surfaces have a central role in numerous scientific and technological fields, including catalysis, thin film growth and tribology. Many applications require knowledge of the strength of these forces as a function of position in three dimensions, but until now such information has only been available from theory. Here, we demonstrate an approach based on atomic force microscopy that can obtain this data, and we use this approach to image the three-dimensional surface force field of graphite. We show force maps with picometre and piconewton resolution that allow a detailed characterization of the interaction between the surface and the tip of the microscope in three dimensions. In these maps, the positions of all atoms are identified, and differences between atoms at inequivalent sites are quantified. The results suggest that the excellent lubrication properties of graphite may be due to a significant localization of the lateral forces.