We present spectroscopic follow-up observations of 68 red, faint candidates from our multi-epoch, multiwavelength, previously published survey of NGC 2264. Using near-infrared spectra from VLT/KMOS, we measure spectral types and extinction for 32 young low-mass sources. We confirm 13 as brown dwarfs in NGC 2264, with spectral types between M6 and M8, corresponding to masses between 0.02 and 0.08 M. These are the first spectroscopically confirmed brown dwarfs in this benchmark cluster. 19 more objects are found to be young M-type stars of NGC 2264 with masses of 0.08–0.3 M. 7 of the confirmed brown dwarfs as well as 15 of the M-stars have IR excess caused by a disc. Comparing with isochrones, the typical age of the confirmed brown dwarfs is <0.5–5 Myr. More than half of the newly identified brown dwarfs and very low-mass stars have ages <0.5 Myr, significantly younger than the bulk of the known cluster population. Based on the success rate of our spectroscopic follow-up, we estimate that NGC 2264 hosts 200–600 brown dwarfs in total (in the given mass range). This would correspond to a star-to-brown dwarf ratio between 2.5:1 and 7.5:1. We determine the slope of the substellar mass function as α = 0.43+0.41−0.56; these values are consistent with those measured for other young clusters. This points to a uniform substellar mass function across all star-forming environments.