The Kepler and K2 missions enabled robust calculations of planet occurrence rates around FGKM-type stars. However, these missions observed too few stars with earlier spectral types to tightly constrain the occurrence rates of planets orbiting hotter stars. Using TESS, we calculate the occurrence rate of small(1R_{Earth}_<R_p_<8R_{Earth}_), close-in (P_orb_<10d) planets orbiting A-type stars for the first time. We search a sample of 20,257 bright (6<T<10) A-type stars for transiting planets using a custom pipeline and vet the detected signals, finding no reliable small planets. We characterize the pipeline completeness using injection-recovery tests and determine the 3{sigma} upper limits of the occurrence rates of close-in sub-Saturns (4R_{Earth}_<R_p_<8R_{Earth}_), sub-Neptunes (2R_{Earth}_<R_p_<4R_{Earth}_), and super-Earths (1R_{Earth}_<R_p_<2R_{Earth}_). We find upper limits of 2.2+/-0.4 sub-Saturns and 9.1+/-1.8 sub-Neptunes per 1000 A-type stars, which may be more than 3x and 6x lower than Kepler-era estimates for Sun-like stars. We calculate an upper limit of 186+/-34 super-Earths per 1000 A-type stars, which may be more than 1.5x lower than that for M dwarfs. Our results hint that small, close-in planets become rarer around early-type stars and that their occurrence rates decrease faster than that of hot Jupiters with increasing host star temperature. We discuss plausible explanations for these trends, including star-disk interactions and enhanced photoevaporation of planet atmospheres.