This is incorrect.
The MINI's power steering system is an electro-hydraulic system. What this means is that rather than an engine driven pump pressurizing the hydraulic fluid as in a conventional p/s system, we have an electric motor doing the job. In all other respects it's a conventional p/s system, it's only the method of pressurizing the fluid that's different.
The p/s system is NOT a sealed system and there IS a reservoir for fluid. It's not mentioned in the manual but it's there. In my Cooper S, the reservoir is immediately adjacent to the coolant expansion tank, not sure if the location is different on a Cooper. It is a small black tank with two relatively thick fluid hoses traveling down the firewall from it to the p/s pump.
I suspect the reason they don't mention it in the manual is because they don't want the owners adding the incorrect p/s fluid and damaging the pump or hydraulic seals. You can however remove the cap and check the fluid level. Attached to the cap is a rudimentary dipstick, just like in conventional p/s systems. There are two lines on the dipstick, neither of which are marked. My Cooper S however has fluid up to the second line when the system is cold and the engine is off. There is additional room inside the reservoir for fluid expansion as it heats up.
It would be wise to check the reservoir and make sure there is fluid inside. If not, you could have a big problem. P/S fluid is not consumed in normal operation so if fluid is missing, it has leaked somewhere, either onto the ground or internally past the seals in the steering rack into the rack itself.
Also the suggestion about checking the p/s cooling fan is also good. If the p/s pump overheats it can cut out.