There have been many debates on the subject. Apple’s iOS automatically capitalizes it, while Android does not. But this isn’t a battle of operating systems.
Here’s what the 2009 AP Stylebook says:
Internet A decentralized, worldwide network of computers that can communicate with each other. In later references, the Net is acceptable.
It goes on to talk about how to use it in stories, because it’s a journalist’s Bible. In any case, the answer is yes - Internet should be capitalized in all uses.
Also, note that it says “the Net” with a capitalization of “Net.” Therefore, when referring to the Internet in this way, one should capitalize “Net.”