The incident took place outside of Ljungby in southern Sweden when the hunter accidentally killed a 71-year-old skier with a single shot that went through the elk.
The hunter was found not guilty of manslaughter, but prosecutor Marcus Sjöstrand will now take the case to the court of appeals.
"I claim that she's been negligent and should be convicted of involuntary manslaughter," Sjöstrand told news agency TT.
A court ruled on Thursday the 32-year-old woman was innocent of all criminal charges, but it was not a unanimous decision.
The court chairman, Åke Belfrage, claimed the woman was guilty as charged, but the other three judges voted not to convict the woman.
Sjöstrand told Sveriges Radio (SR) that it's important to try this case in the court of appeals, and that he believes in a conviction at that level.
Torgny Eklund, who represented the relatives of the deceased, didn't want to comment on their reactions following the verdict.
"We will go through this verdict step by step when it has sunk in a little," Eklund told the Sveriges Radio.
"We are between Christmas and New Year's Eve, so there are some other things to deal with as well."
The 32-year-old hunter had held her license for six years when her first elk was felled in December 2010 with a single shot.
But just 60 metres beyond the felled beast lay a 71-year-old cross-country skier in the snow.
The bullet which killed the elk had contined, hitting the skier and killing him instantly.