Dendrite growth threatens the safety of batteries by piercing the ion-transporting separators between the cathode and anode. Finding a dendrite-suppressing material that combines high modulus and high ionic conductance has long been considered a major technological and materials science challenge. Here we demonstrate that these properties can be attained in a composite made from Kevlar-derived aramid nanofibres assembled in a layer-by-layer manner with poly(ethylene oxide). Importantly, the porosity of the membranes is smaller than the growth area of the dendrites so that aramid nanofibres eliminate 'weak links' where the dendrites pierce the membranes. The aramid nanofibre network suppresses poly(ethylene oxide) crystallization detrimental for ion transport, giving a composite that exhibits high modulus, ionic conductivity, flexibility, ion flux rates and thermal stability. Successful suppression of hard copper dendrites by the composite ion conductor at extreme discharge conditions is demonstrated, thereby providing a new approach for the materials engineering of solid ion conductors.