It's a matter of style.
The Guardian style guide uses lowercase:
net, web, world wide web
web, webpage, website, world wide web
is there at least some consensus among the more popular style guides/dictionaries on this?
Not really. Wikipedia has a page on the topic, Internet capitalization conventions, with a section on usage:
Examples of media publications and news outlets that capitalize the term include The New York Times, the Associated Press, Time, and The Times of India. In addition, many peer-reviewed journals and professional publications such as Communications of the ACM capitalize "Internet", and this style guideline is also specified by the American Psychological Association in its electronic media spelling guide.
More recently, a significant number of publications have switched to not capitalizing the noun "internet." Among them are The Economist, the Financial Times, The Times, the Guardian, the Observer and the Sydney Morning Herald. As of 2011, most publications using "internet" appear to be located outside of North America, but the gap is closing. Wired News, an American news source, adopted the lower-case spelling in 2004. Around April 2010, CNN shifted its house style to adopt the lowercase spelling.
Wired has a 2004 piece on their decision to use internet, web and net.
Pick your own style and be consistent.