Nanoparticle superlattices are hybrid materials composed of close-packed inorganic particles separated by short organic spacers. Most work so far has concentrated on the unique electronic, optical and magnetic behaviour of these systems. Here, we demonstrate that they also possess remarkable mechanical properties. We focus on two-dimensional arrays of close-packed nanoparticles and show that they can be stretched across micrometre-size holes. The resulting free-standing monolayer membranes extend over hundreds of particle diameters without crosslinking of the ligands or further embedding in polymer. To characterize the membranes we measured elastic properties with force microscopy and determined the array structure using transmission electron microscopy. For dodecanethiol-ligated 6-nm-diameter gold nanocrystal monolayers, we find a Young's modulus of the order of several GPa. This remarkable strength is coupled with high flexibility, enabling the membranes to bend easily while draping over edges. The arrays remain intact and able to withstand tensile stresses up to temperatures around 370 K. The purely elastic response of these ultrathin membranes, coupled with exceptional robustness and resilience at high temperatures should make them excellent candidates for a wide range of sensor applications.