The self-assembly of organic molecules into supramolecular materials with structural ordering beyond the nanometre scale is challenging. Here, we report the spontaneous self-assembly of a chiral discotic triphenylene derivative into millimetresized droplets. The structure of the droplets is characterized by high positional and orientational ordering and a three-dimensional integrity similar to that of single crystals. Notwithstanding, these assemblies slide when placed on a vertical substrate demonstrating their fluid nature. X-ray imaging shows that during the sliding process the internal crystal-like structure is maintained and that the droplets undergo clockwise or counterclockwise unidirectional rotation, depending on the chirality of their molecular components. Rheological measurements suggest that this rotational behaviour might result from the distinct yield stress between the (R)- and (S)-enantiomers. Overall, our findings demonstrate that molecular chirality can determine the movement direction of a supramolecular structure, thus expanding the fundamental understanding of the structure and dynamics of soft materials.