Fast is one of a category of adjectives that double as adverbs without requiring the -ly ending. Because fast works this way—and it does for all its main senses, including in a speedy manner and in a secure manner or tightly—the adjectival fastly is a superfluous word. The -ly adds nothing.
There is some resistance to flat adverbs (that is, adverbs such as fast that take adjectival forms), and when using fast this way you might encounter people who claim the word should be changed to quickly or some other -ly synonym. This is out of step with modern English usage, though, and it’s the sort of useless English rule that should be laid to rest.
In these sentences, fast would work in place of fastly:
[W]ith the Feb. 2nd signing day fastly approaching, I am sure we will be talking a lot more. [Nation of Blue]
Until now, those fastly evolving cities did not have any StreetView coverage and it proved difficult for people to imagine what they look like for real. [press release via GISUser.com]
Currently, shooting of the film is progressing fastly. [16 Reels (link now dead)]
And in these instances, fast is an adverb:
But when the Bellenden Road space came free, she moved fast. [Evening Standard]
Lucy talks fast, thinks fast, and observes, digests, and personalizes visual trends fast. [In Fashion, Annemarie Iverson]
And I don’t just mean physical structures; our societal structures have held fast as well. [Los Angeles Times]
When you tighten the screw, give it an extra turn to be sure the wire is secured fast to the screw. [Do it Right the First Time, Gene Hamilton and Kate Hamilton]