We report on current-induced magnetization switching in a nanomagnet with perpendicular anisotropy, and investigate the effects of the damping constant (α) on the switching current (Isw) by varying the nanosecond-scale pulse current duration (tp), the saturation magnetization (Ms), and the magnetocrystalline anisotropy (Ku). The results show that reduction of α below a certain threshold (αc) is ineffective in reducing Isw for short tp. When tp is short, it is necessary to reduce both α and Ms simultaneously until αc is reached to reduce Isw. The results presented here offer a promising route for the design of ultrafast information storage and logic devices using current-induced magnetization switching.