Yes, since they are the only ones processing the custom transaction related to Splinterlands from the Hive blockchain they could block them.
I mean to clarify and say currently the way they ban is by not allowing banned users to access the official Splinterlands website.
I don't believe they have ever banned someone from the API. Even though from a technological standpoint they could, I don't believe they do that and it would be a large change in the way they operate/enforce policy if they did.
Is there a source that actually confirms this? Seems... unlikely to be done that way from a technical standpoint. As a software engineer, it makes zero sense to me that they would ban an account from a website ui and not just interacting with the server as a whole.