To investigate possible variations in the stellar initial mass function (IMF) in red-sequence galaxies, we have obtained infrared spectroscopy with Subaru/Fibre Multi-Object Spectrograph (FMOS) for a sample of 92 red-sequence galaxies in the Coma cluster. Velocity dispersions, ages and element abundances for these galaxies have been previously determined from optical data. The full range of velocity dispersions covered by the sample is σ = 50–300 . By stacking the FMOS spectra in the rest frame and removing sky-subtraction residuals and other artefacts fixed in the observed frame, we derive composite spectra in the 9600–10500 Å range for galaxies grouped according to their velocity dispersion or Mg/Fe ratio. We measure the Wing–Ford band of FeH and a new index centred on a Ca i line at 10345 Å; these features are strong in cool dwarf stars, and hence reflect the form of the IMF at low mass (<0.5 M⊙). The Ca i line, unlike the Wing–Ford band and other ‘classical’ IMF indicators (Na i doublet and Ca ii triplet), is unaffected by the abundance of sodium. We compare the measured indices against predictions from spectral synthesis models matched to the element abundances estimated from the optical data. Binning galaxies by velocity dispersion, we find that both the Wing–Ford and Ca i index measurements are best reproduced by models with the Salpeter IMF. There is no clear evidence for an increase in dwarf-star content with velocity dispersion over the range probed by our sample (which includes few galaxies at the highest velocity dispersions, σ > 250 ). Binning the observed galaxies instead by Mg/Fe ratio, the behaviour of both indices implies a trend of IMF from Chabrier-like, at abundance ratios close to solar, to Salpeter or heavier for highly α-enhanced populations. At face value, this suggests that the IMF depends on the mode of star formation, with intense rapid starbursts generating a larger population of low-mass stars.