Close inspection showed that it was likely to be a broken wire inside the white cable, probably just by the microphone (the white cable looked thin and weak at this point - it had probably been bent to-and-fro too much).
This instructable shows some pictures of the steps needed to repair it.
Don't bother trying, unless you have:
- a fine soldering iron
- a sharp knife
- a strong magnifying glass
- a very fine pair of tweezers
- a steady hand.
Step 1: Find the problem
Step 2: Open up the microphone
I took a very sharp knife and gently worked my way along the seam, going part-way through at first, then going back along. Eventually, the corners started to part and some leverage began to open up the plastic case.
The side with the buttons, came off first, leaving the tiny printed circuit board (PCB) wedged (glued?) into the other half of the case.
Sliding the knife-point under the PCB made it pop out.