  We present J-band spectroscopy of passive galaxies focusing on the Na I
doublet at 1.14 {\mu}m. Like the Na I 0.82 {\mu}m doublet, this feature is
strong in low-mass stars and hence may provide a useful probe of the initial
mass function (IMF). From high signal-to-noise composite spectra, we find that
Na I 1.14 {\mu}m increases steeply with increasing velocity dispersion,
{\sigma}, and for the most massive galaxies (\sigma > 300 km/s) is much
stronger than predicted from synthetic spectra with Milky-Way-like IMFs and
solar abundances. Reproducing Na I 1.14 {\mu}m at high {\sigma} likely requires
either a very high [Na/H], or a bottom-heavy IMF, or a combination of both.
Using the Na D line to break the degeneracy between IMF and abundance, we infer
[Na/H] $\approx$ +0.5 and a steep IMF (single-slope-equivalent x $\approx$ 3.2,
where x = 2.35 for Salpeter), for the high-\sigma galaxies. At lower mass
({\sigma} = 50-100 km/s), the line strengths are compatible with MW-like IMFs
and near-solar [Na/H]. We highlight two galaxies in our sample where strong
gravitational lensing masses favour MW-like IMFs. Like the high-{\sigma} sample
on average, these galaxies have strong Na I 1.14 \mu m; taken in isolation
their sodium indices imply bottom-heavy IMFs which are hard to reconcile with
the lensing masses. An alternative full-spectrum-fitting approach, applied to
the high-\sigma sample, recovers an IMF less heavy than Salpeter, but
under-predicts the Na I 1.14 \mu m line at the 5{\sigma} level. We conclude
that current models struggle to reproduce this feature in the most massive
galaxies without breaking other constraints, and caution against over-reliance
on the sodium lines in spectroscopic IMF studies.
