Differentially rotating hypermassive neutron stars (HMNSs) are metastable objects which can be formed in the merger of neutron-star binaries. The eventual collapse of a HMNS into a black hole is a key element in generating the physical conditions that are expected to accompany the launch of a short gamma-ray burst. We investigate the influence of magnetic fields on HMNSs by performing global three-dimensional simulations in general-relativistic magnetohydrodynamics (MHD). In particular, we provide direct evidence for the occurrence of the magnetorotational instability (MRI) in HMNS interiors. For the first time in simulations of these systems, rapidly-growing and spatially-periodic structures are observed to form, with features like those of the channel flows produced by the MRI in other systems. Moreover, the growth time and wavelength of the fastest-growing mode can be extracted and compared successfully with analytical predictions. The MRI emerges as an important mechanism to amplify magnetic fields over the lifetime of the HMNS, whose collapse is found to be accelerated by MHD effects. The evidence provided here that the MRI can actually occur in HMNSs could have a profound impact on the outcome of neutron-star binary mergers and their connection to short gamma-ray bursts.