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Recently, evidence for a conducting surface state (CSS) below 19 K was reported for the correlatedd-electron small gap semiconductor FeSi. In the work reported herein, the CSS and the bulk phase of FeSi were probed via electrical resistivity ρ measurements as a function of temperatureT, magnetic fieldBto 60 T, and pressurePto 7.6 GPa, and by means of a magnetic field-modulated microwave spectroscopy (MFMMS) technique. The properties of FeSi were also compared with those of the Kondo insulator SmB₆to address the question of whether FeSi is ad-electron analogue of anf-electron Kondo insulator and, in addition, a “topological Kondo insulator” (TKI). The overall behavior of the magnetoresistance of FeSi at temperatures above and below the onset temperatureT_S= 19 K of the CSS is similar to that of SmB₆. The two energy gaps, inferred from the ρ(T) data in the semiconducting regime, increase with pressure up to about 7 GPa, followed by a drop which coincides with a sharp suppression ofT_S. Several studies of ρ(T) under pressure on SmB₆reveal behavior similar to that of FeSi in which the two energy gaps vanish at a critical pressure near the pressure at whichT_Svanishes, although the energy gaps in SmB₆initially decrease with pressure, whereas in FeSi they increase with pressure. The MFMMS measurements showed a sharp feature atT_S≈ 19 K for FeSi, which could be due to ferromagnetic ordering of the CSS. However, no such feature was observed atT_S≈ 4.5 K for SmB₆.