Combinations of high strength and ductility are hard to attain in metals. Exceptions include materials exhibiting twinning-induced plasticity. To understand how the strength-ductility trade-off can be defeated, we apply in situ, and aberration-corrected scanning, transmission electron microscopy to examine deformation mechanisms in the medium-entropy alloy CrCoNi that exhibits one of the highest combinations of strength, ductility and toughness on record. Ab initio modelling suggests that it has negative stacking-fault energy at 0K and high propensity for twinning. With deformation we find that a three-dimensional (3D) hierarchical twin network forms from the activation of three twinning systems. This serves a dual function: conventional twin-boundary (TB) strengthening from blockage of dislocations impinging on TBs, coupled with the 3D twin network which offers pathways for dislocation glide along, and cross-slip between, intersecting TB-matrix interfaces. The stable twin architecture is not disrupted by interfacial dislocation glide, serving as a continuous source of strength, ductility and toughness.