Interband cascade infrared photodetectors (ICIPs) combine interband optical transitions with fast intraband transport to achieve high-frequency and broad-wavelength operation at room temperature. Here we study the bias-dependent electronic impulse response of ICIPs with a mid-infrared synchronously pumped optical parametric oscillator (OPO). Since the OPO produces ultrashort 104-fs pulses, it is possible to probe the impulse response of the ICIP. From this impulse response, we identify two characteristic decay times, indicating the contribution of electron as well as hole carriers. A reverse bias voltage applied to the ICIP reduces both time scales and leads to an increased electrical cut-off frequency. The OPO emits up to 500 mW average power, of which up to 10 mW is directed to the ICIP in order to test its saturation characteristics under short-pulse illumination. The peak of the impulse response profile as well as the average photocurrent experience a gradual saturation behavior, and we determine the corresponding saturation powers by measuring the photo-response as a function of average power directed to the ICIP. We demonstrate that an increasing reverse bias increases the saturation power as well as the responsivity of the ICIP.