I have/had the same on my 9070 shifters. While I have no explanation (nor Shimano does), I solved it adding a drop of silicone sealant. Process looks like:
- Disconnect the shifter wire
- Press the noisy lever (buttons paddle) and put something that keeps it depressed.
- Add a drop of silicone sealant and let it cure overnight (make sure temperature is high enough per silicone specs)
- When dry, remove whatever you put to keep lever depressed.
The dried silicone will act as rubber seal/silencer and the paddle won't make any noise.
Not that I'm happy about a DIY solution for a bloody expensive system. For some reason I'd expect Shimano Di2 to work flawlessly and making noise is indeed a flaw.
Note: another poster suggested to add some grease to the internal spring. I considered this also and it makes a lot of sense. Just be prepared for take the shifter out the handlebar -not absolute requirement but recommended for better accessibility- and deal with torx5 screws and small pieces. And, please... do not lose any of those screws/small pieces