Context. The cosmic infrared background (CIB) consists of emission from distant, dusty, star-forming galaxies. Energetically, the CIB is very important as it contains as much energy as the extragalactic optical background. The nature and evolutionary status of the objects making up the background are, however, unclear. Aims. The CIB peaks at ∼150 µm, and as such is most effectively studied from space. The limited apertures of space-borne telescopes set the angular resolution that can be attained, and so even Herschel, with its 3.5 m diameter, will be confusion-limited at this wavelength at ∼5 mJy. The bulk of the galaxies contributing to the CIB are fainter than this, so it is difficult to study them without interferometry. Here we present the results of a preliminary study of an alternative way of probing fainter than the continuum confusion limit using far-IR imaging spectroscopy. An instrument capable of such observations is being planned for SPICA -a proposed Japanese mission with an aperture equivalent to that of Herschel and more than 2 orders of magnitude more sensitive. Methods. In this paper we investigate the potential of imaging spectrometers to break the continuum confusion limit. We have simulated the capabilities of a spectrometer with modest field of view (2 × 2 ), moderate spectral resolution (R ∼ 1-2000) and high sensitivity. Results. We find that such an instrument is capable of not only detecting line emission from sources with continuum fluxes substantially below the confusion limit, but also of determining their redshifts and, where multiple lines are detected, some emission line diagnostics. Conclusions. 3-D imaging spectrometers on cooled far-IR space telescopes will be powerful new tools for extragalactic far-IR astronomy.