ANNECY-LE-VIEUX, France — The list of endearing quirks connected to the Iceland men’s soccer team is maddeningly, frustratingly, delightfully long. Try to pick a favorite: Is it the coach who is also a dentist? Is it the announcer whose television calls sound as if he has been lighted on fire? Is it the 10 percent — and no, that is not an exaggeration — of Iceland’s residents who have come to France to cheer for their team at this summer’s European Championships?

All are worthy contenders — as is, it should be said, the goalkeeper who used to direct music videos — but regardless of which absurdity strikes anyone’s particular fancy, the sum of these parts has produced one of the more remarkable international sports stories in recent memory.

Iceland, whose national population of about 330,000 makes it smaller than more than 50 American cities, qualified for the Euros (which was startling), advanced out of the group stage (which was surprising), beat England in the round of 16 (which was astonishing) and now, in Sunday’s quarterfinal at the national stadium just outside Paris, will try to eliminate the host, France (which would be, frankly, ridiculous).

“We are living a dream,” said Gudmundur Benediktsson, a former national team player who does play-by-play for the team’s television broadcasts. “The whole country is living a dream. And we don’t want to wake up.”