Indeed the country has already had a female president, Vigdis Finnbogadottir, who was the world's first democratically elected female head of state in 1980.
"It might sound big news in other countries," says President Olafur Ragnar Grimsson who refuses to comment on the campaign per se, or on his rival, "but so far, we've achieved a state in Iceland where it's not particularly remarkable that a woman holds high office."
Indeed, Icelanders aren't particularly in awe of Ms Arnorsdottir for being a young mother of three with a newborn baby.
Her supporters back her because she is the first viable contender to Mr Grimsson in nearly two decades.
"I watch my children and I just think I want to do what I can to influence the society they're growing up in," Ms Arnorsdottir says.
"Which values we are building, what kind of atmosphere there is. Talking with the elderly, they say 'You've got three kids? I had 10!'.
"I've always worked hard and even though this job is being a president, it doesn't change anything. I think the challenge is a matter of getting used to the unusual idea."
If Ms Arnorsdottir wins, her 42-year-old partner will become a house husband. He will relinquish his career, also as a television journalist (he and Ms Arnorsdottir met on location).
"They tell me I'm a role model," Mr Halldorsson says as he expertly holds the baby while Ms Arnorsdottir meets people in an old people's home on another campaign stop.
"I'm happy with that. It doesn't make me a lesser man. I can change diapers and still watch the football, cook and fix the car."
On her 17-hour day campaigning outside Reykjavik, Ms Arnorsdottir drives past the Snaefellsnes Glacier, renowned for its mystical powers.
People living nearby claim it is one of the earth's few energy sources. If she were looking out of the minibus window, Ms Arnorsdottir may have been hoping for some of that mystical power to help trump her rival and some of the energy to do the job if she succeeds at the polls on Saturday.