We present a detailed study of the Na I line strength index centred in the K band at 22 100 Å (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-the-art scaled-solar stellar population models, even using our newly developed models in the near infrared (NIR). The models clearly underestimate the large NaI2.21 values measured for most early-type galaxies. However, we develop an 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.