We analyse a robust sample of 30 near-infrared-faint (KAB> 25.3, 5σ) submillimetre galaxies (SMGs) selected from a 0.96deg2 field to investigate their properties and the cause oftheir faintness in optical/near-infrared wavebands. Our analysisexploits precise identifications based on Atacama Large Millimeter Array(ALMA) 870-μm continuum imaging, combined with very deepnear-infrared imaging from the UKIDSS Ultra Deep Survey. We estimatethat SMGs with KAB > 25.3 mag represent 15 ± 2 percent of the total population brighter than S870 = 3.6 mJy,with a potential surface density of ∼450 deg-2 aboveS870 ≥ 1 mJy. As such, they pose a source of contaminationin surveys for both high-redshift 'quiescent' galaxies and very highredshift Lyman-break galaxies. We show that these K-faint SMGs representthe tail of the broader submillimetre population, with comparable dustand stellar masses to KAB ≤ 25.3 mag SMGs, but lying atsignificantly 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 investigatethe origin of the strong dust attenuation and find indications thatthese K-faint galaxies have smaller dust continuum sizes than theKAB ≤ 25.3 mag galaxies, as measured by ALMA, whichsuggests their high attenuation is related to their compact sizes. Weidentify a correlation of dust attenuation with star formation ratesurface density (ΣSFR), with the K-faint SMGsrepresenting the higher ΣSFR and highest AVgalaxies. The concentrated, intense star formation activity in thesesystems is likely to be associated with the formation of spheroids incompact galaxies at high redshifts, but as a result of their highobscuration these galaxies are completely missed in ultraviolet,optical, and even near-infrared surveys.