Running on (i.e. when you switch off the ignition but the engine continues to run - usually rather roughly - for anything up to 5-10 seconds) is quite common on older Minis especially 1275's with high compression ratios
The most common causes are using too low a grade of petrol, the wrong type of spark plug, incorrect ignition timing, too high an idle speed, excessive build-up of carbon in the combustion chambers or running too hot. What is happening is that when you switch off the ignition, a spark plug, valve or piece of carbon continues to glow red hot, igniting any remaining petrol/air mixture.
The following will sometimes avoid it happening: Making sure the engine is in an optimum state of tune, letting it idle for a few seconds before you switch it off after a long fast run, or flooring the throttle immediately after you've switched off the ignition to weaken any remaining fuel/air mixture. Physically stalling the engine by engaging first gear and letting the clutch out sharply should only he considered as a last resort!
Most recent cars including Rover Minis, now feature a device to prevent it happening. The pre-injection Mini with its variable jet SU carburettor uses a solenoid controlled valve (energize to close) which opens to allow air into the inlet manifold. The idea is that when you switch off the ignition the valve opens and the sudden influx of a large quantity of air makes any petrol/air mixture so weak that it cannot ignite.
To fit an anti-run-on valve to older Minis is quite straightforward. All you need is the late Mini Cooper solenoid valve itself, which can be bought from a UK or NZ Rover dealer. Alternatively, valves can be found in wreckers yards (if you know someone in the UK) on many recent cars such as MG Metro, etc. It should be mounted vertically or within 30 degrees of vertical with the gauze covered intake trumpet uppermost. On the bulkhead is ideal away from heat, water and vibration. Vacuum-resistant servo hose should be used to connect it either direct to the manifold via a suitable connection drilled and tapped in the manifold, or simply by inserting a 'T' or 'Y' piece in an existing brake servo hose if the car has one.
Electrical connections are simply a feed off the fusebox or coil which is live only when the engine is running (i.e. ignition controlled) and an earth connection to the body.
Loss of Oil Pressure During Hard Cornering
Does your Mini suffer from loss of oil pressure when you go to fast around right hand corners, especially roundabouts? This problem is caused by all the oil draining to the left side of the sump and away from the oil pickup which is located on the right hand side. The resulting induction of air into the lubrication system can result in an engine staved of oil for several seconds.
The solution to this problem is to fit a central oil pickup pipe, which costs about \$65 and is available from most Mini parts suppliers. However, fitting the oil pickup pipe is not straight forward due to the fact that the normal oil pickup can't be completely removed without dismantling the gearbox. So if you're getting your gearbox rebuilt then its worthwhile getting a central oil pickup installed as an extra bit of insurance for your engine.