We present K-band photometry and low-resolution near-infrared spectroscopy from 1.44 to 2.45 μm of isolated planetary-mass candidate members in the σ Orionis cluster found by Zapatero Osorio et al. The new data have been obtained with NIRC at the Keck I telescope. All of our targets, except for one, are confirmed as likely cluster members. Hence, we also confirm that the planetary-mass domain in the cluster is well populated. Using our deep K-band images, we searched for companions to the targets in the separation range of 0.3"-10" up to a maximum faint limit of K=19.5 mag. One suspected companion seems to be an extremely red galaxy. The near-infrared colors of the σ Orionis substellar members indicate that dust grains condense and settle in their atmospheres. We estimate that the surface temperatures range from 2500 down to 1500 K. The spectroscopic sequence covers the full range of L subclasses, and the faintest object is tentatively classified as T0. These targets provide a sequence of substellar objects of known age, distance, and metallicity, which can be used as a benchmark for understanding the spectral properties of ultracool dwarfs.