I created a bug with Microsoft here, but after debugging with them it appeared that the issue was with the Oracle Extension. They suggested opening a ticket so that the root cause could be explored:
[CoPilot hangs when trying to create a .sql file · Issue #276564 · microsoft/vscode](https://github.com/microsoft/vscode/issues/276564)
Once it hangs, VS Code has to be restarted for Copilot to become responsive again