The periodic activity of repeating fast radio burst (FRB) 180916.J0158+65 was recently reported by the CHIME/FRB Collaboration team. 28 bursts from this source not only show a $sim16$-day period with an active phase of $sim 4.0$ days but also exhibit a broken power law in differential energy distribution. In this paper, we suggest that FRB 180916.J0158+65-like periodic FRBs would provide a unique probe of extragalactic asteroid belts (EABs), based on our previously-proposed pulsar-EAB impact model, in which repeating FRBs arise from an old-aged, slowly-spinning, moderately-magnetized pulsar traveling through an EAB around another stellar-mass object. These two objects form a binary and thus the observed period is in fact the orbital period. We show that this model can be used to well interpret all of the observed data of FRB 180916.J0158+65. Furthermore, we constrain the EABs physical properties and find that (1) the outer radius of the EAB is at least an order of magnitude smaller than that of its analogue in the solar system, (2) the differential size distribution of the EABs asteroids at small diameters (large diameters) is shallower (steeper) than that of solar-system small objects, and (3) the two belts have a comparable mass.
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