The concept of an Interplanetary_Internet is gradually becoming possible.
https://www.datacenterdynamics.com/en/analysis/building-internet-moon/
https://qz.com/1919694/the-moon-is-getting-4g-internet-before-4-billion-earthlings/
https://www.theverge.com/22393952/glass-fiber-internet-cable-wifi-speed-5g-explainer-video
https://canteltelecom.com/2020/01/14/5-reasons-that-make-fiber-optic-future-proof/
https://marspedia.org/Internet#Concept_for_a_Martian_Internet
Of course basic communication conduits had to first be established on earth before ever thinking of the moon and planets...
https://en.wikipedia.org/wiki/Submarine_communications_cable#Antarctica
The_Victorian_Internet era was a good attempt in trying to connect more of the world. Today, billions of people are still without having an Internet access device.
https://www.computerhistory.org/revolution/networking/19/406
https://tomstandage.wordpress.com/books/the-victorian-internet/
https://www.goodreads.com/book/show/52853.The_Victorian_Internet
https://upload.wikimedia.org/wikipedia/commons/thumb/7/7d/World_map_of_submarine_cables.png/640px-World_map_of_submarine_cables.png
Global submarine cables in 2007
https://en.wikipedia.org/wiki/Global_network
https://en.wikipedia.org/wiki/Submarine_communications_cable#Importance_of_submarine_cables
https://kaw.wallenberg.org/en/research/broadband-mars-new-optical-amplifier
https://www.nasa.gov/connect/apps.html
https://www.zdnet.com/article/this-is-how-nasa-chooses-tech-for-the-international-space-station-and-why-ai-could-help-get-us-to Mars and beyond.
https://www.pcmag.com/news/4-android-smartphones-with-as-much-power-as-nasas-mars-helicopter
https://www.theverge.com/tldr/2021/3/2/22309412/nasa-perseverance-mars-rover-processor-cpu-imac-1998
https://science.nasa.gov/science-news/science-at-nasa/2015/31oct_tablet
http://physicsbuzz.physicscentral.com/2010/01/mars-tablets-and-hydrogel.html
Of course the initial startup of the Interplanetary_Internet will be a gradual step by step process.
https://en.wikipedia.org/wiki/Interplanetary_Internet#Protocol
https://en.wikipedia.org/wiki/Interplanetary_Internet#Earth_orbit
https://en.wikipedia.org/wiki/Intergalactic_Computer_Network
https://en.wikipedia.org/wiki/Galactic_Empire_(Isaac_Asimov)
https://en.wikipedia.org/wiki/Foundation_series
https://en.wikipedia.org/wiki/Foundation_(TV_series)
While the moon is an important steppingstone, Mars remains as the first likely planet for initial interplanetary AI and human colonization.