"Flirting with Pride and Prejudice" is definitely fanfic. There's a lot of published fanfic on that novel. Doesn't mean it is necessarily bad though, some of the fanfic online is actually quite good. People can be as snobby about fanfic as they are about comic books and soap operas, highly amusing considering the things they love which fall in those categories. ;-)

Some award-winning fanfic based on characters/novels in the public domain include:

The Wide Sargasso Sea - features Rochester's insane wife and their love story
March - features Mr. March of Louisa May Alcott's Little Women and his story.
Ahab's Wife - based on an unseen character in Moby Dick

Real-person fanfic:

Marilyn by Joyce Carol Oats...