Update: it seems that 2.1.139 is the culprit, whereas 2.1.138 worked fine!
They have instructions on installing and running older versions: https://code.claude.com/docs/en/setup#install-a-specific-version
But you probably don't want to do that over security concerns.
I have also create a GitHub bug report: https://github.com/anthropics/claude-code/issues/59062
If they don't fix it, I just will have to do that in the next version of ccode, by running a local proxy.