It is known that graphene exhibits natural ripples with characteristic lengths of around 10 nm. But when it is stretched across nanometre-scale trenches that form in a reconstructed copper surface, it develops even tighter corrugations that cannot be explained by continuum theory. Understanding how the mechanical behaviour of materials deviates at the nanoscale from the macroscopically established concepts is a key challenge of particular importance for graphene, given the complex interplay between its nanoscale morphology and electronic properties1,2,3,4,5. In this work, the (sub)nanometre-wavelength periodic rippling of suspended graphene nanomembranes has been realized by thermal strain engineering and investigated using scanning tunnelling microscopy. This allows us to explore the rippling of a crystalline membrane with wavelengths comparable to its lattice constant. The observed nanorippling mode violates the predictions of the continuum model6, and evidences the breakdown of the plate idealization7 of the graphene monolayer. Nevertheless, microscopic simulations based on a quantum mechanical description of the chemical binding accurately describe the observed rippling mode and elucidate the origin of the continuum model breakdown. Spatially resolved tunnelling spectroscopy measurements indicate a substantial influence of the nanoripples on the local electronic structure of graphene and reveal the formation of one-dimensional electronic superlattices.