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.