Carrier depletion and transport in a single ZnO nanowire Schottky device have been investigated at 5 K, using cathodoluminescence measurements. An exciton diffusion length of 200 nm has been determined along the nanowire axis. The depletion width is found to increase linearly with the reverse bias. The origin of this unusual dependence in semiconductor material is discussed in terms of charge location and dimensional effects on the screening of the junction electric field.