AutoB,
When was the last time this worked? What has been done since then?
Looks like you have a permanent earth connection.
Mine is a RHD car but I suspect that the light is the same. This is how it connects and works on my car.
From the drivers seat the switch is to the left side of the mirror and the bulb on the right. A no brainer really, the actual workings will only fit in in this way.
I have a purple wire going to the fuse box and a black wire going to the door switch. The black wire connects to switch side of the assembly, the purple to the bulb side of the assembly. I think it its important that this is correct.
(For clarification I think my purple wire is your white wire and the in reality the wires plug into a little plastic plate that sits in the top of the mirror frame.)
With the switch flicked to the left the light stays on all the time (door opened or closed) Here the earth is coming throught the metal mirror frame directly.
With the switch flicked to the right the light stays on when the door is opened, but goes off when the door is closed. This should be the position the switch is in by default. Here the earth is controlled from the door switch.
If your wiring at the light looks OK then I suspect that the door switch may be an issue. Perhaps you should unscrew it and have a look.
Mine was rusted so badly there was no bright metal to make a good earth contact. Yours may be broken and fused together so you get a permanent earth connection.
Be careful when you unscrew the switch in the door. If you are not careful and you pull the wire off the back and it falls inside the door jamb it is a bit of a bugger to fish it out. It can be done with a piece of wire with a hook on the end of it and a torch.
Regards
Joe R