The work function of palladium is known to be sensitive to hydrogen by the formation of a surface dipole layer or Pd hydride. One approach to detect such a change in the work function can be based on the formation of a Schottky barrier between the palladium metal and a semiconductor. Here, we study the hydrogen sensitivity of Schottky barrier field-effect transistors made for the first time from diameter- and chirality-sorted semiconducting single-walled carbon nanotubes (s-SWNTs) in contact with Pd electrodes. We observe an unrivaled 100-fold change in the on-state conductance at 100 ppm H2 compared to air for devices with s-SWNT and diameters between 1 and 1.6 nm. Hydrogen sensing is not observed for devices of Pd-contacted few-layer graphene (FLG), as expected due to the absence of a significant Schottky barrier. Unexpectedly, we observe also a vanishing sensitivity for small-diameter SWNTs. We explain this observation by changes in the nanotube work function caused by spillover and chemisorption of atomic hydrogen onto small-diameter nanotubes. We also observe that long-term sensing stability is only achieved if the gate voltage is inverted periodically. Under constant gate bias, the sensitivity reduces with time, which we relate to gate screening by accumulated charges in the substrate.