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It is generally recognized that very narrow silicon-on-insulator (SOI) fin field-effect transistors (FinFETs) are insensitive to substrate bias due to the strong electrostatic gate control. In this letter, we demonstrate, for the first time, that, in short-channel narrow FinFETs, substrate bias can dramatically change the on-current without change in the threshold voltage, subthreshold slope, and drain-induced barrier lowering, due to the modulation of the parasitic series resistance. Therefrom, contrary to general belief, very narrow short-channel multiple-gate field-effect transistors can be sensitive to substrate-related effects (buried oxide formation, irradiation, etc). Another important implication of the described effect is related to the diagnostics of the series resistance in SOI FinFETs and better prediction of their full intrinsic performance potential.