Recently there is a flourishing and notable interest in the crystalline scintillator material sodium iodide (NaI) as target for direct dark matter searches. This is mainly driven by the long-reigning contradicting situation in the dark matter sector: the positive evidence for the detection of a dark matter modulation signal claimed by the DAMA/LIBRA collaboration is (under so-called standard assumptions) inconsistent with the null-results reported by most of the other direct dark matter experiments. We present the results of a first prototype detector using a new experimental approach in comparison to conventional single-channel NaI scintillation light detectors: a NaI crystal operated as a scintillating calorimeter at milli-Kelvin temperatures simultaneously providing a phonon (heat) plus scintillation light signal and particle discrimination on an event-by-event basis. We evaluate energy resolution, energy threshold and further performance parameters of this prototype detector developed within the COSINUS R&D project.