  We present a detailed study of the Na I line strength index centered in the
$K$-band at $22100$, {\AA} (NaI2.21 hereafter) relying on different samples of
early-type galaxies. Consistent with previous studies, we find that the
observed line strength indices cannot be fit by state-of-art scaled-solar
stellar population models, even using our newly developed models in the NIR.
The models clearly underestimate the large NaI2.21 values measured for most
early-type galaxies. However, we develop a Na-enhanced version of our newly
developed models in the NIR, which - together with the effect of a bottom-heavy
initial mass function - yield NaI2.21 indices in the range of the observations.
Therefore, we suggest a scenario in which the combined effect of [Na/Fe]
enhancement and a bottom-heavy initial mass function are mainly responsible for
the large NaI2.21 indices observed for most early-type galaxies. To a smaller
extent, also [C/Fe] enhancement might contribute to the large observed NaI2.21
values.
