Option to make browser extension window remain open and pinned (always on top)

No, not that I’m aware of, but here is a work-around that you may or may not like:

  1. Make sure that your browser extension is locked.
  2. Open a new browser tab in your browser.
  3. Use the autofill shortcut Ctrl+Shift+L.
  4. Unlock your browser extension.

The above process works in Chrome; I have not tested it on other browsers.