A name ending in .base.eth is a subname issued by Base. It is not managed in the ENS App, and ENS support cannot change it for you.
Where to manage it
Base calls these names Basenames and runs them at base.org/names. That is where you renew one, edit its records, and transfer it.
The ENS App manages .eth names registered on Ethereum Mainnet. A .base.eth name is a different thing with a different owner, even though the two look similar.
Why it still resolves in ENS apps
Base issues its names underneath base.eth, which is itself an ENS name. That is why wallets and apps can read a .base.eth name and show it like any other. Reading it and managing it are separate things: anyone can read it, only Base can manage it.
Common questions
Who do I contact about my .base.eth name?
Base, at base.org/names. We have no control over their names and cannot see them, renew them, or move them.
Can I use it as my primary name?
Yes. From the moment you register it, it is already your primary name on Base. Base sets that, and you can change it. You can also make it your Ethereum primary name in the ENS App. It cannot be set at primary.ens.domains.
I want a .eth name instead. What do I do?
Register one in the ENS App. It is a separate registration, and holding a .base.eth name does not reserve the matching .eth name for you. See Register a .eth name for the steps.