Myredzed has done a fantastic write up for the non-///Ms, but the procedure of removal/refit is different for the ///M variants, hence this post. I hope some of you find it useful.
It seems a common problem for the aluminium bases on our mirrors to oxidise, causing unsightly bubbling on an otherwise immaculate car;
I tackled this over the last few days, without too much bother at all. You just need time and patience.
You will need;
1. Allen Keys
2. Torx keys
3. An aerosol can of matched colour (Halfords can make one up for you for about £12)
4. An aerosol can of acid etch primer (normal primer won't key to the aluminium as well as this stuff)
5. An aerosol can of clear lacquer
5. Posidriv Screwdriver
6. Flathead Screwdrivers
7. Masking Tape
8. Long Nosed Pliers
Step 1 - Glass Removal
This can be a bit fiddly, and requires you to push the top of the mirror in as far as it will go, so you can just about see underneath the bottom of the glass and into the guts of the mirror body.
You the need to take a small flathead driver and try to ease the black clips of the mirror itself, off from the white mirror bracket. You will only be able to do the bottom 2;
This picture shows the black clips to identify, and how they are constructed;
This picture shows the mirror bracket you are trying to pop them off from - (note the 4 white studs the clips attach to);
Once you have popped the bottom two off, get your hand behind the glass, and using appropriate force, try to pull the glass off from the top two. This will require some wriggling, but I could not access the top clips at all in the same manner I reached the bottom - the mirror simply does not move far in enough at the bottom to allow you get a screwdriver into the top.
Once the glass is off, simply pull the two wires of the heated glass off. Use some long nose pliers if need be, and put the glass aside safe.
Now the glass is off, you will see this;
Undo the four screws (posidriv), this will release the motor mechanism and allow you access to the back of the motor unit.
There are four terminals in the back - each wire can simply be pulled out using long nose pliers, and you don't need to remember where they go, as the instructions of what-goes-where, are moulded onto the back of the motor unit!
Once these have been removed, put the motor aside.
Feed the heated mirror cables back into the stalk - this will make the wires less bulky and easier to pull through from the other side
Step 2 - Body Removal
Pull out mirror toward you (away from the car) and twist.
Passenger side you pull out and turn clockwise
Driver's side you pull out and turn anti clockwise
This will reveal two hex head (Allen head) bolts which simply unscrew.
NOTE; To reduce the chance of the mirror breaking when trying to twist it (which Jonttt informed me some people have reported), ensure you pull the mirror away from the car by about 4/5mm BEFORE twisting. The mirror body is spring loaded tight to the base, and must be compressed by pulling toward you before rotating

Pull the body away from the car, pull off the black rubber base, and gently pull the wires through;
Note the two heated mirror cable coming through to make it easier to pull the rest through...
The body is now free!
For protection against the elements (as the painting takes days), I cleaned up the exposed wires, and then taped them together neatly. This will also make it easier to thread them back into the mirror upon reattachment.
NOTE; Be sure to put a little insulation tape over each metal crimp though, else they will short if you start the car up.
I then covered the hole in the door with a plastic bag and taped it to the side of the car using masking tape, as snow was forecast;
Whilst all the innards of the mirror were out, I cleaned them all up using G101;
There are a couple of door components there too - ignore them!
Step 3 - Prepping The Bases Ready For Paint
You cannot actually separate the base from the mirror body, as the internal bracket will not fit out of the neck... (The ordinary Z3 mirrors are two-piece, so I believe they can be taken apart). On the ///M mirrors, we must just undo the 4 torx bolts that fix the bracket to the body, and then feed the bracket as far out through the neck as it can go. I found turning it 90 degrees allowed it come out the furthest.
Once in a suitable position, tape the bracket in place, to stop it from moving around;
These were taken at the end, when all painted and finished, but you can see how I did it;
Once all bound in place with tape, mask off everything else except the base (I put a freezer bag over each body and taped the rest up), ready for prep.
Using some heavy grit abrasive paper, I attacked the worst of the oxidisation, and then once I got down to the metal, I used varying degrees of wet and dry (200, 400, 800) until I had a smooth transition between paint and metal.
Make sure you get rid of every single area of oxidised aluminium (the white milky stuff), else it'll come back.
De-grease the surface, and the allow it to dry thoroughly - again, any trapped moisture will allow the bubbling to return.
Once thoroughly degreased and dry, you can get on with the painting.
Step 4 - Paint
You must use acid etch primer to allow a good key to the aluminium.
Do one good layer of primer, and allow to dry thoroughly for a good 8 hours - in front of a heater would be even better.
Then use very fine (800 and 1200 grit) wet and dry paper with water to smooth the primed surface - the smoother you make each layer of paint, the better the end result will be.
Once done, dry thoroughly and repeat with a second coat of primer. Again, allow to dry bone dry, so you can wet and dry it smooth once more - this time make sure it has a very smooth finish, as paint is next.
After smoothing;
OK. Now the tricky bit, where you will make or break the final finish - Paint!
What works best for me is a light, constant spray. Have a few trial sprays with the can, just to get used to the feel of the paint can and nozzle.
Once you've nailed the fine spray, whilst holding the mirror in one hand make passes with the spray can, whilst rotating the mirror to ensure all surfaces are evenly covered.
Don't spray on too heavy, else the paint will, run, drip and get air bubbles in it - if this happens, you will have to leave it to dry thoroughly for a day, just so you can sand it back down and start again.
Once you have done one coat, allow to harden overnight.
Lightly sand with very fine wet and dry (1200/1500) using water to lubricate. This will be the last time you sand the surface, so what you do here will determine the final finish. Degrease and dry thoroughly.
Now for the second and final coat of paint. Repeat the step above with regard to spraying - light spray, nice and even, make sure everything is covered.
Allow to dry in front of a gentle heater, or overnight.
Final stage; lacquer. This is much easier than paint, but the same rules apply as if it were paint - make it nice and even, covering everything.
Repeat for a second coat, allow to dry overnight - this was the final coat, and must be left to dry thoroughly, else handling it too early may leave fingerprints in the soft bottom layers of paint/lacquer.
You should be left with an immaculate finish like this;
Happy?

Now, remove all masking and if you want, apply a coat of wax.
Realign the bracket in the mirror body, and replace the four torx screws.
Step 5 - Refitting Mirrors
Same as reversal.
1. Feed cables through the rubber base and into the mirror.
2. Fit rubber base to the bottom of the mirror, align, and tighten the two hex bolts attaching the mirror to the car nice and tight.
3. Pull mirror outward, and swivel back to the original position.
4. Re-terminate the wires into the motor unit using the long nose pliers - colours are indicated on each terminal, so you can't get it wrong.
5. Screw motor unit onto body (4 screws).
6. Attach heated mirror wires.
7. Push mirror back onto clips (can be tricky, but persist).
Polish everything up, and admire your 'as-new' bases.

Note the snow in the background... I cleaned the car with hot water before I reassembled, as snow was totally covering the car! Was absolutely frozen by the end of it all!
Hope you have found this guide helpful, and saved a few quid doing it yourself.
Keith.