This work reports the nonlinear dynamics of a mid-infrared interband cascade laser (ICL) subject to optical injection. It is shown that the stable locking regime is asymmetric and broadens with increasing injection strength. Outside the locking regime, the ICL mostly produces period-one oscillations. However, three categories of periodic pulse oscillations are observed in the vicinity of the Hopf bifurcation and the saddle-node bifurcation. In particular, it is found that the ICL generates broadband chaos at a near-threshold pump current, and the chaos bandwidth is over 300 MHz.