I have published an app in Google Play in Production mode. Now, I have a new version which I want to be release in Beta mode for my limited number of private beta users. Is it possible to have them both? i.e. version 1.0 in production mode and version 1.1 in beta mode? Or should I maintain a different app for beta (which is not convenient, as I need to change package names).
you can have both, technically you can have 4 apks available to users and depending on other requirements can have even more than that.
Alpha - alpha testers who opt in can test it
Beta - beta testers who opt in can test, can be different from alpha
Staged Rollout - percentage of users can download
Production - all users, or latest version
https://support.google.com/googleplay/android-developer/answer/3131213?hl=en