ABSTRACT We analyse a robust sample of 30 near-infrared-faint (KAB > 25.3, 5σ) submillimetre galaxies (SMGs) selected from a 0.96 deg2 field to investigate their properties and the cause of their faintness in optical/near-infrared wavebands. Our analysis exploits precise identifications based on Atacama Large Millimeter Array (ALMA) 870-μm continuum imaging, combined with very deep near-infrared imaging from the UKIDSS Ultra Deep Survey. We estimate that SMGs with KAB > 25.3 mag represent 15 ± 2 per cent of the total population brighter than S870 = 3.6 mJy, with a potential surface density of ∼450 deg−2 above S870 ≥ 1 mJy. As such, they pose a source of contamination in surveys for both high-redshift ‘quiescent’ galaxies and very high redshift Lyman-break galaxies. We show that these K-faint SMGs represent the tail of the broader submillimetre population, with comparable dust and stellar masses to KAB ≤ 25.3 mag SMGs, but lying at significantly higher redshifts (z = 3.44 ± 0.06 versus z = 2.36 ± 0.11) and having higher dust attenuation (AV = 5.2 ± 0.3 versus AV = 2.9 ± 0.1). We investigate the origin of the strong dust attenuation and find indications that these K-faint galaxies have smaller dust continuum sizes than the KAB ≤ 25.3 mag galaxies, as measured by ALMA, which suggests their high attenuation is related to their compact sizes. We identify a correlation of dust attenuation with star formation rate surface density (ΣSFR), with the K-faint SMGs representing the higher ΣSFR and highest AV galaxies. The concentrated, intense star formation activity in these systems is likely to be associated with the formation of spheroids in compact galaxies at high redshifts, but as a result of their high obscuration these galaxies are completely missed in ultraviolet, optical, and even near-infrared surveys.