so the power steering pumps on gen 1 mini ie 2001-2006 for hatchback and convertibles up to 2008 have a pump that fails in soooo many ways from bearing failures to seals failing etc,, there is a way that it does fail and drains the battery as its control circuit board does one and never switches off, this can also put heavy load on to the alternator which is possible to overload that as well, even seen one put so much load on v=battery it finished it off,
hence why its never worth guessing whats wrong with them ie need a multi meter and read a few things, first being battery voltage with car running on full charged battery 14.2-14.7v, the battery after being charged over night let stand for 20 minutes then read across it, good battery 12.8v bad battery 12.2v and under, then connect battery to car and try and start it keep multimeter on battery your looking for a perfect battery stays above 12.6v and worn out battery will dump its voltage fairly fast, its sort of bush way of working it out, also car running on idle fully charged battery turn all lights on heater fan heated windows etc everything on, the voltage should stay above 13.9v ,, the other way involves expensive test gear, above way is just a cost of £5 meter off ebay