Conventional composite materials offer high specific stiffness and strength, but suffer from low failure strains and failure without warning. This work proposes a new design for sandwich structures with symmetrically-wavy composite skins and a crushable foam core, aiming to achieve large strains (due to unfolding of the skins) and energy absorption (due to crushing of the foam core) under tensile loading. The structure is designed by a combination of analytical modelling and finite element simulations, and the concept is demonstrated experimentally. When loaded under quasi-static tension, wavy-ply sandwich specimens with carbon–epoxy skins and optimised geometry exhibited an average failure strain of 8.6%, a specific energy dissipated of 9.4 kJ/kg, and ultimate strength of 1570 MPa. The scope for further developing the wavy-ply sandwich concept and potential applications requiring large deformations and energy absorption are discussed.