Latest official update [2023-01-09]:

We are currently working on Py3.10 support (see #31). Py3.11 will ship separately. We don't currently have a timeline to share on Py3.11 support. We will we post updates on this issue when we have further information to share.

And another official update from a different thread [2023-02-28]:

We don't have a ETA to share as yet.

Python 3.11 will be was released on 2022-10-24 (the final release candidate was released on 2022-09-12)

• Announcements:
  • Release Candidate 2: https://pythoninsider.blogspot.com/2022/09/python-3110rc2-is-now-available.html
  • 3.11.0: https://pythoninsider.blogspot.com/2022/10/python-3110-is-now-available.html
  • 3.11.1: https://pythoninsider.blogspot.com/2022/12/python-3111-3109-3916-3816-3716-and.html
  • 3.11.2: https://pythoninsider.blogspot.com/2023/02/python-3112-python-31010-and-3120-alpha.html
  • 3.11.3: https://pythoninsider.blogspot.com/2023/04/its-time-for-another-set-of-python.html
• Release schedule: https://peps.python.org/pep-0664/
• What’s new in Python 3.11: https://docs.python.org/3.11/whatsnew/3.11.html

(I know that 3.10 isn't available yet on Lambda; but AWS could add both at the same time and not let us wait for > 1 year until they add 3.11 🙏)