In theory it should be possible, but I never tried it. Many users had problems with (unintentionally) running two versions, which were mainly caused by one installation being in a local user directory. Should you decide to try it, be sure to install the new version in a different folder somewhere in "Program Files". In addition, consider creating a copy of the whole message base, pointing the new version to that, and telling the new TB version to leave messages on the server. Of course, make regular back-ups of your message base anyway.

Personally, I would always install the new version over the old one, and never had the need to go back. If you purchased your v9 license after June 1, 2021, it should work with v10 as well. In any case, RitLabs has reduced the upgrade prices significantly since the last time I purchased one (by ~50%).

Finally, make sure to use the newest v10.3.3.3, not the Christmas edition v10.3.3 (one "3" less) which stopped working a few days ago.

What I always do is make a copy of the folder c:\Program Files\The Bat!\ and call it something like "c:\Program Files\The Bat!v8.8.9". Also, as my MAIL folder is located elsewhere (on a different drive) I make a copy of that as well, like "D:\The Bat!v8.8.9\MAIL".

Then I install the new version of TheBat on top of the current one so everything is upgraded correctly. Of course, as noted above "and telling the new TB version to leave messages on the server" is vitally important. Its not an issue for me because I always have that setting turned on. I only delete messages when I want to delete messages.

Then if I want to go back, just rename the two directories.