We derive a new map of the interstellar extinction near the Galactic Centre (GC), extending to much higher values of AV than previously available, and use the results obtained to better characterise the long-period variable star population of the region. We take the Spitzer IRAC catalogue of GC point sources (Ramírez et al. 2008) and combine it with new isochrones (Marigo et al. 2008) to derive extinctions based on the photometry of red giants and asymptotic giant branch (AGB) stars. We apply it to deredden the LPVs found by Glass et al. (2001) near the GC (Glass-LPVs). We make period-magnitude diagrams and compare them to those from other regions of different metallicity. Our new extinction map of the GC region covers 2.0 DEG * 1.4 DEG (280 * 200 pc at a distance of 8 kpc). The Glass-LPVs follow well-defined period-luminosity relations (PL) in the IRAC filter bands at 3.6, 4.5, 5.8, and 8.0 micron . The period-luminosity relations are similar to those in the Large Magellanic Cloud, suggesting that the PL relation in the IRAC bands is universal. We use ISOGAL data to derive mass-loss rates and find for the Glass-LPV sample some correlation between mass loss and pulsation period, as expected theoretically. Theoretical isochrones for a grid of different metallicities and ages are able to reproduce this relation. The GC has an excess of high-luminosity and long-period LPVs compared to the bulge, which supports previous suggestions that it contains a younger stellar population.