I expect that there are quite a few stories that predict an internet-like system. One quite well-known short story is 'The Machine Stops' by E.M. Forster published in 1909 and widely reprinted in science-fiction anthologies. It concerns a future Earth where everyone lives underground in cells and never meet except through the Machine where they discuss art, culture etc. Their entire life is essentially conducted on-line. It all depends how specific you want to define 'internet-like'. William Gibson is generally credited as being the first to write about the internet in the way we know it now, but he had probably already learned of it in its first military incarnation. There are probably other, less well-known instances stretching back to H G Wells. So, I think the whole thing is a bit of a myth.

I also read one time that no-one had predicted mobile phones in science fiction. That seems unlikely as well. The concept of using telephones that are portable must have occurred to writers at the dawning of radio. It seems inconceivable that no-one envisaged that kind of technology.