Upgrading EventStore in Production

Is it possible to upgrade an eventstore cluster that is running in production one node at a time (rolling upgrade)? Or does the whole database cluster need to come down in order to upgrade all database nodes together?

You can upgrade as you have described. Take down one node, upgrade and bring that node back up.

Providing same major version … If there are any kind of breaking changes there will be a version increase, then check release notes.