Yep.
And last time i checked, Tasmania was part of Australia.
Not sure how long Flynn spent in Australia during his life, though.
M. K. Gandhi would win the most famous Indian title hands down. Mother Teresa would give him some competition as the most famous person associated with India.
Other possible famous Indians: Ismail Merchant (of the Merchant-Ivory duo), Zubin Mehta (conductor), Freddy Mercury, Vinod Khosla (to geeks, I guess)…
I imagine Don Bradman is a known entity on the Indian subcontinent, which is more populous that the US. Ian Thorpe is a tourism ambassador, which has taken him on tour in China.
I believe Russell Crowe has Australian citizenship, FWIW.
Or perhaps Yoko Ono?
For Hong Kong, I guess it would be Jackie Chan. Bruce Lee if we’re counting associations.
Can it be extended to fiction? Because then you open the doors to Superman, Robin Hood, King Arthur et al.
Denmark. H.C.Anderson, Kierkegaard or Niels Bohr, Svend Forkbeard.
Norway/Iceland: Eric the Red, Leif the Happy.
In my case, I think it’s either Julius Caesar or Augustus.
He is in fact one of the most popular entertainers in China. He should also be rich by now but I’m not certain of that.
You mean Rome or Italy? How about Michelangelo, Leonardo de Vinci, Marco Polo, Galilei, Columbus, Aquinas – and I think both Nero and Caligula are better known than Augustus. Italy seem to have more famous persons than any other country.
Err…So Sicilians aren’t Italians, for instance?
Apparently, islands don’t count, in this thread, for some reason…
Tasmania / Australia is fairly obvious, but that Corsica is French is at least debatable. Even though it is so legally today, it wasn’t so for much of history, and what percentage of Corsicans consider themselves French today. Didn’t Napoleon’s family hold fervent anti-French sentiments?
Though I agree Napoleon should be considered French and one of the greates Europeans of all time.
Right now I’m in Qatar. There are not, and never have been, any famous Qataris.
First person I think of is Parvo Nurmi. After all his nickname was The Flying Finn. He was responsible for possibly the greatest mile race of all time at the '24 Olympics when he won the gold medal in the mile and the 5000 metres within 45 minutes of one another.
Other famous Finns:
Linus Torvalds (linux guy)
Mika Häkkinen, Mika Salo and Kimi Räikkonen (Fornula 1 drivers)
and most famous of all:
Huckleberry Finn 
I remember hearing somewhere that Muhammad Ali is the world’s most recognizable person – I tried to google this and found plenty of mentions of this “fact,” but no information on how this was arrived at. Still, it makes some sense, when you consider his popularity in the non-Western world. George Washington never went to Africa.
Although this claim is a bit dated, I’m still going to put him forward as the most famous American.
A bit sketchy this, but some would argue you can claim Santa.
kellner, not obvious to me, really. 
Do you mean the guy who would beat Arnie to it if we forget the “associated with” rule and go for country of birth only?
For my native Holland, a bunch of painters:
Vermeer, Van Gogh (the painter, not the filmmaker) and Rembrandt.
Jjjimmm covered Ireland well enough.
Both Rome (well, the Roman Republic) and Italy. I’m under the impression that many, many people heard, at the very least, about Julius Caesar and some of his exploits. Of those you mention, I think only Leonardo da Vinci could compete, and then again I met people who think he’s French…
Gibraltar: tiny (pop. 29,000) but has produced a couple of minor celebs:
Albert Hammond: you might not know him by name but you’ll recognise the songs:
It never rains in southern California
He ain’t heavy he’s my brother
When I need you
and others. His son Albert Hammond jr. is in The Strokes.
John Galliano, fashion designer.
Iceland, I bet, would be Bjork, hands down.