Yes!
The Ambire UI is open-source, can be cloned from GitHub and ran locally.
Some of the features require a connection to our relayer, but you do not need that in order to access your funds.
For email/password accounts, because the second key is on the relayer side, if the relayer goes down, you'll either need a backup of your second key (which you can request to be exported) or you'll need to wait 3 days for the account recovery procedure to reset your second key.