Free-standing nanostructures such as suspended carbon nanotubes, graphene layers, III-V nanorod photonic crystals and three-dimensional structures have recently attracted attention because they could form the basis of devices with unique electronic, optoelectronic and electromechanical characteristics. Here we report the growth by molecular beam epitaxy of free-standing nanoplates of InAs that are close to being atomically plane. The structural and transport properties of these semiconducting nanoplates have been examined with scanning electron microscopy, transmission electron microscopy, X-ray diffraction and low-temperature electron transport measurements. The carrier density of the nanoplates can be reduced to zero by applying a voltage to a nearby gate electrode, creating a new type of suspended quantum well that can be used to explore low-dimensional electron transport. The electronic and optical properties of such systems also make them potentially attractive for photovoltaic and sensing applications.