A mathematical model is developed to describe the characteristic behavior of a small vibro-impact pile driver. Dynamics of the small vibro-impact pile driver is represented by a three-dimensional map. The map is of piecewise property due to synchronous and non-synchronous motion of the driver and pile immediately after the plastic impact, and singularities caused by grazing contact of the driver and pile. The pile driver exhibits two different types of single-impact periodic motions in different regions of the forcing frequency due to the plastic impacts. Transition of two types of single-impact periodic motions is demonstrated, and the influence of the piecewise property, singularities and various parameters on the performance of the pile driver is analyzed.