We analyse a robust sample of 30 near-infrared-faint (KAB > 25.3, 5σ) submillimetre galaxies 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 ALMA 870-μm continuum imaging, combined with very deep near-infrared imaging from the UKIDSS-UDS survey. We estimate that submillimetre galaxies 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 submillimetre galaxies represent the tail of the broader submillimetre population, with comparable dust and stellar masses to KAB ≤ 25.3 mag submillimetre galaxies, 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, that 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 submillimetre galaxies 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 UV, optical and even near-infrared surveys.