What will the version of the Internet that follows us to the stars look like? Yes, people are really thinking seriously about this sort of thing. Specifically the InterPlanetary Networking Special Interest Group (IPNSIG).
Ansible-like faster-than-light communication it isn't. There's no magical warp drive. Nor is a network of telepaths acting as a 'verse spanning telegraph system.
It's more mundane than that. And in many ways more interesting as it's sort of like the old Internet on steroids, the one that was based on on UUCP and dial-up connections, but over vastly longer distances and with much longer delays: