We present far-infrared (FIR) and submillimetre photometry from the Herschel Space Observatory's Spectral and Photometric Imaging Receiver (SPIRE) for 313 nearby (z<0.05) active galactic nuclei (AGN). We selected AGN from the 58 month Swift Burst Alert Telescope (BAT) catalogue, the result of an all-sky survey in the 14-195keV energy band, allowing for a reduction in AGN selection effects due to obscuration and host galaxy contamination. We find 46 per cent (143/313) of our sample is detected at all three wavebands and combined with our Photoconductor Array Camera and Spectrometer (PACS) observations represents the most complete FIR spectral energy distributions of local, moderate-luminosity AGN. We find no correlation among the 250, 350, and 500{mu}m luminosities with 14-195keV luminosity, indicating the bulk of the FIR emission is not related to the AGN. However, Seyfert 1s do show a very weak correlation with X-ray luminosity compared to Seyfert 2s and we discuss possible explanations. We compare the SPIRE colours (F_250_/F_350_ and F_350_/F_500_) to a sample of normal star-forming galaxies, finding the two samples are statistically similar, especially after matching in stellar mass. But a colour-colour plot reveals a fraction of the Herschel-BAT AGN are displaced from the normal star-forming galaxies due to excess 500{mu}m emission (E_500_). Our analysis shows E_500_ is strongly correlated with the 14-195keV luminosity and 3.4/4.6{mu}m flux ratio, evidence the excess is related to the AGN. We speculate these sources are experiencing millimetre excess emission originating in the corona of the accretion disc.