The distribution of elastic strains (and thus stresses) at the submicrometre length scale within deformed metal single crystals has remarkably broad implications for our understanding of important physical phenomena. These include the evolution of the complex dislocation structures that govern mechanical behaviour within individual grains, the transport of dislocations through such structures, changes in mechanical properties that occur during reverse loading (for example, sheet-metal forming and fatigue), and the analyses of diffraction line profiles for microstructural studies of these phenomena. We present the first direct, spatially resolved measurements of the elastic strains within individual dislocation cells in copper single crystals deformed in tension and compression along <001> axes. Broad distributions of elastic strains are found, with important implications for theories of dislocation structure evolution, dislocation transport, and the extraction of dislocation parameters from X-ray line profiles.