In this study, a compressive impact test was conducted using the split Hopkinson pressure bar (SHPB) method to investigate SMA fiber-reinforced mortar’s impact behavior. A 1.5% fiber volume of crimped fibers and dog-bone-shaped fibers was used, and half of the specimens were heated to induce recovery stress. The results showed that the appearance of SMA fibers, recovery stress, and composite capacity can increase strain rate. For mechanical properties, the SMA fibers reduced dynamic compressive strength and increased the peak strain. The specific energy absorption of the reinforced specimens slightly increased due to the addition of SMA fibers and the recovery stress; however, the effect was not significant. The composite behavior between SMA fibers and the mortar matrix, however, significantly influenced the dynamic compressive properties. The higher composite capacity of the SMA fibers produced lower dynamic compressive strength, higher peak strain, and higher specific energy absorption. The composite behavior of the dog-bone-shaped fiber was less than that of the crimped fiber and was reduced due to heating, while that of the crimped fiber was not. The mechanical properties of the impacted specimen followed a linear function of strain rate ranging from 10 to 17 s−1; at the higher strain rates of about 49–67 s−1, the linear functions disappeared. The elastic modulus of the specimen was independent of the strain rate, but it was dependent on the correlation between the elastic moduli of the SMA fibers and the mortar matrix.