An interplanetary internet does not exist yet. When people talk about an interplanetary internet, they are talking about a problem they are still trying to solve - the problem of making the internet to work between different planets.
The way the internet works here on Earth is simple. Computers need to connect to each and can share a network. One computer sends another computer a message (called a packet) and then the other sends back a message saying it got it. This works fine because the messages only take a second to get back and forth. But if you were sending that message to Mars, it would take about ten minutes for the message to get there, and another ten for it to get back. That means we need to completely change the way computers talk to each other if we are going to communicate between different planets. That is the problem people are trying to solve when they talk about the interplanetary internet. It's a new and a very interesting concept.