Should you capitalize “Internet”?

Yes, according to Alexander v. Verizon Wireless Services LLC:

Although many style guides, such as the Chicago Manual of Style, and news sources, such as the Associated Press, no longer instruct writers to capitalize “Internet,” we decline to follow this trend. For many, such as the New York Times, the reason for the change to “internet” is simple: others were doing it, so they thought they should, too.  “Internet,” however, was originally capitalized to distinguish the global network from other internets—short for “inter networks”—which are collections of smaller networks that communicate using the same protocols.  In our view, this still makes the word a proper noun, regardless of how often people refer to other internets. Furthermore, to the extent “decapitalizing [I]nternet is part of a universal linguistic tendency to reduce the amount of effort required to produce and process commonly-used words,” we reject the tasks of striking an additional key or reading over a capital “I” as persuasive reasons to alter a word.

No. 16-31227 n.12 (Nov. 13, 2017) (citations omitted).