Inverse Compton Scattering (ICS) is an emerging compact X-ray source technology, where the small source size and high spectral brightness are of interest for multitude of applications. However, to satisfy the practical flux requirements, a high-repetition-rate ICS system needs to be developed. To this end, this paper reports the experimental demonstration of a high peak brightness ICS source operating in a burst mode at 40 MHz. A pulse train interaction has been achieved by recirculating a picosecond CO2 laser pulse inside an active optical cavity synchronized to the electron beam. The pulse train ICS performance has been characterized at 5- and 15- pulses per train and compared to a single pulse operation under the same operating conditions. With the observed near-linear X-ray photon yield gain due to recirculation, as well as noticeably higher operational reliability, the burst-mode ICS offers a great potential for practical scalability towards high duty cycles.