We present an analysis of Spitzer IRAC (3.6-8um) and MIPS (24um) imaging of members of the 16^+10^_-5_ Myr old open cluster NGC 1960 (M36). Models of terrestrial planet formation indicate that rocky planets are likely to achieve their final masses at around 10-30Myr, and thus this cluster is at an interesting epoch for planet formation. We find 21 B-F5 type stars and 14 F6-K9 type stars which have 24um excess emission, and thus determine that >30 per cent of B-F5 type stars and >23 per cent of F6-K9 type stars in this cluster have 24um excess emission. These excess frequencies are similar to those observed in other clusters of similar age. Three early-type stars have excesses at near-infrared wavelengths. Analysis of their spectral energy distributions confirms that these are true debris discs and not remnant primordial or transitional discs. None of the 61 Sun-like stars has confirmed near-infrared excess, and we can place a limit on the frequency of 8um excess emission around Sun-like stars of <7 per cent. All of the detected excesses are consistent with emission from debris discs and are not primordial.