We present millimetre (Submillimeter Array) and submillimetre (SCUBA-2) continuum observations of the peculiar star KIC 8462852 which displayed several deep and aperiodic dips in brightness during the Kepler mission. Our observations are approximately confusion-limited at 850 ?m and are the deepest millimetre and submillimetre photometry of the star that has yet been carried out. No significant emission is detected towards KIC 8462852. We determine upper limits for dust between a few 10-6 M? and 10-3 M? for regions identified as the most likely to host occluding dust clumps and a total overall dust budget of for regions identified as the most likely to host occluding dust clumps and a total overall dust budget of ? within a radius of 200 au. Such low limits for the inner system make the catastrophic planetary disruption hypothesis unlikely. Integrating over the Kepler light curve we determine that at least 10-9 M? of dust is required to cause the observed Q16 dip. This is consistent with the currently most favoured cometary breakup hypothesis, but nevertheless implies the complete breakup of ~30 Comet 1/P Halley type objects. Finally, in the wide SCUBA-2 field of view we identify another candidate debris disc system that is potentially the largest yet discovered.
