The pics above are similar to the issues on mine except mine have burnt into the laquer rather than onto the surface - if surface hard rainwater marks the vinegar wash should work - acid, alkali and all of that.
I've had proper advice now - it seems as if the view is that the paint was of incorrect quality and / or incorrectly applied leaving it too soft, and allowing damage from outside factors that wouldn't normally damage paintwork.
As far as I know soft paint (the car's 3 months old) through incorrect application or incorrect mixing can't really be fixed - baking it again may not work, and besides, the sun has done that already a bit this summer.
The fact that BMW are seemingly doing sweet FA is the bit that's driving me mad. The car's going in shortly and a BMW technical manager is going to assess it and make a suggestion on how to fix it.
Can you imagine what bird lime would do to this paint, or tree sap!
I'll let you know how I get on so I can advice other people on an appropriate route of action.....
Finally - being in the UK not familiar with clay - presumably this is a light abrasive compund which moves surface blemishes?