High-entropy alloys, near-equiatomic solid solutions of five or more elements, represent a new strategy for the design of materials with properties superior to those of conventional alloys. However, their phase space remains constrained, with transition metal high-entropy alloys exhibiting only face- or body-centered cubic structures. Here, we report the high-pressure synthesis of a hexagonal close-packed phase of the prototypical high-entropy alloy CrMnFeCoNi. This martensitic transformation begins at 14 GPa and is attributed to suppression of the local magnetic moments, destabilizing the initial fcc structure. Similar to fcc-to-hcp transformations in Al and the noble gases, the transformation is sluggish, occurring over a range of >40 GPa. However, the behaviour of CrMnFeCoNi is unique in that the hcp phase is retained following decompression to ambient pressure, yielding metastable fcc-hcp mixtures. This demonstrates a means of tuning the structures and properties of high-entropy alloys in a manner not achievable by conventional processing techniques.