Recently, another miracle discovery was made in the highlands by Lesja, not far from the area in question, where a Viking Age sword of amazing condition had spent over a thousand years in a glacier, owing its condition to a combination of sub-zero temperatures, and a sheltered, well ventilated spot between the rocks. Despite this, it's got nothing on the from Vik in terms of conservation.
Though they might exist, I am not aware of any examples quite like it. There are however other artifacts which have been ascribed a similar provenance. One famous case being the so-called King Olaf's Helmet, a relic taken from Trondheim to Sweden in 1564, during the Northern Seven Years' War. In reality, it never belonged to the martyred king, and neither is it viking helmet, but a 15th century sallet, forged some 450 years after king Olaf's death.
In the humorous 19th century travelogue Three in Norway (by two of them), one of the two English narrators mention a visit to the farm Bjølstad in Heidal, which is right up the road from Vik. There their host Ivar Tofte (amiably nicknamed Bluebeard) presented them with a similar curiosity during a tour of the estate:
Bluebeard first took us through the state apartments, which contained many curious and interesting things of all ages, from an axe nearly a thousand years old, to a Birmingham plated teapot won at the Christiania horse show in 1860.
Of course, we don't know whether or not "Bluebeard" was correct: Knowledge about viking weapon typology was still crude in the 1880's, nearly 40 years before Jan Peteren published his dissertation on the chronology of Norwegian viking swords. As the story of the helmet of Saint Olaf proves; people without the help of modern archaeology couldn't be trusted to identify an object less than a hundred years old, let alone a thousand.
It's doubtful that a farmer would have been able to determine the age of a viking axe without consulting a specialist, which could also have been the case. Ivar Tofte seems to have been well connected, and was apparently somewhat of a local bigman. However, he could also have relied on local legend and hearsay when he claimed it was "almost a thousand years old". The authors go on to mention that he claimed descendance from Harold Fairhair, another important king, and ancestor of Saint Olaf.
It's speculative, but it makes me wonder if the more well-off farmers around the Gudbrandsdalen valley kept such artifacts to give credence to their claims of legendary ancestry. Demonstrating ties to the Norse past was very fashionable at the time, after all, with Norwegian identity going through a veritable renaissance. This was a time where formerly obscure pieces of Norse literature now being translated and widely distributed. Whatever the fate of the axe: No such artifact from Bjølstad is mentioned in the public record.