We analyse a robust sample of 30 near-infrared-faint (K>25.3, 5 sigma) submillimetre galaxies selected across a 0.96 deg^2 field, to investigate their properties and the cause of their lack of detectable optical/near-infrared emission. Our analysis exploits precise identifications based on ALMA 870um continuum imaging, combined with the very deep near-infrared imaging from the UKIDSS-UDS survey. We estimate that K>25.3 submillimetre galaxies represent 15+/-2 per cent of the total population brighter than S870=3.6mJy, with an expected surface density of ~450/deg^2 above S870>1mJy. 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 are simply the tail of the broader submillimetre population, with comparable dust and stellar masses to K<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 K<25.3 galaxies, as measured by ALMA, which suggests their high attenuation is related to their compact sizes. We find a correlation of dust attenuation with star-formation rate surface density (Sigma_SFR), with the K-faint submillimetre galaxies representing the higher-Sigma_SFR 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 are completely missed in UV, optical and even near-infrared surveys.
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