Nanoscale structures can be highly strained because of confinement effects and the strong influence of their external boundaries. This results in dramatically different electronic, magnetic and optical material properties of considerable utility. Third-generation synchrotron-based coherent X-ray diffraction has emerged as a non-destructive tool for three-dimensional (3D) imaging of strain and defects in crystals that are smaller than the coherence volume, typically a few cubic micrometres, of the available beams that have sufficient flux to reveal the material's structure. Until now, measurements have been possible only at a single Bragg point of a given crystal because of the limited ability to maintain alignment; it has therefore been possible to determine only one component of displacement and not the full strain tensor. Here we report key advances in our fabrication and experimental techniques, which have enabled diffraction patterns to be obtained from six Bragg reflections of the same ZnO nanocrystal for the first time. All three Cartesian components of the ion displacement field, and in turn the full nine-component strain tensor, have thereby been imaged in three dimensions.