The heist was the first ever bank robbery in living memory in the territory, which is located in the Arctic Ocean, about halfway between continental Norway and the North Pole.
"There was an armed robbery at around 10:40 am (0940 GMT)," Terje Carlsen, a spokesman for the local governor, told AFP.
"A man with a gun seized a sum of money. He was arrested quite quickly" in the centre of Longyearbyen, the capital, he added.
Police said the suspected robber was a foreigner travelling in the region who was sent to the northern town of Tromso on the mainland where he will be questioned.
Authorities declined to give more details about the suspect's identity, the amount stolen or the weapon used in the robbery.
The odds of the heist succeeding were always low on the archipelago, famous for glaciers and its polar bears who outnumber residents.
Longyearbyen has around 2,000 inhabitants and practically everyone knows each other. The airport is the main means of leaving the settlement.
Commentators on social media were quick to make fun of the failed bank robbery.
"The most reckless bank robbery in Norwegian history?" said one Twitter user.
"He probably forgot to think about his escape route," commented another.
The Svalbard archipelago, roughly twice the size of Belgium, lies about 1,000 kilometres (600 miles) from the North Pole.
Temperatures in winter regularly plunge to below minus 20 degrees Celsius (minus 4 Fahrenheit) and can drop below minus 40.
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