Ferromagnetic semiconductors (FMSs), which have the properties and functionalities of both semiconductors and ferromagnets, provide fascinating opportunities for basic research in condensed matter physics and device applications. Over the past two decades, however, intensive studies on various FMS materials, inspired by the influential mean-field Zener (MFZ) model have failed to realise reliable FMSs that have a high Curie temperature (Tc > 300 K), good compatibility with semiconductor electronics, and characteristics superior to those of their non-magnetic host semiconductors. Here, we demonstrate a new n type Fe-doped narrow-gap III-V FMS, (In,Fe)Sb, in which ferromagnetic order is induced by electron carriers, and its Tc is unexpectedly high, reaching ~335 K at a modest Fe concentration of 16%. Furthermore, we show that by utilizing the large anomalous Hall effect of (In,Fe)Sb at room temperature, it is possible to obtain a Hall sensor with a very high sensitivity that surpasses that of the best commercially available InSb Hall sensor devices. Our results reveal a new design rule of FMSs that is not expected from the conventional MFZ model. (This work was presented at the JSAP Spring meeting, presentation No. E15a-501-2: this https URL)