The sheep was mauled on the afternoon of March 3 on moorland north of Danby, and its injuries were so severe it had to be put down.
A spokesperson for North Yorkshire Police said: “Attacks on livestock have a huge financial impact, not to mention the distress that a farmer goes through by finding dead or dying animals suffering from horrific injuries.
“It is vital for the safety and welfare of livestock and dogs themselves, that owners keep their pets under control at all times. Failure to do so can have devastating consequences.”
In 2018, North Yorkshire Police contributed to an expert report by the National Police Chiefs’ Council Livestock Worrying Police Working Group. Their research found that about one in ten of livestock attacks involve repeat offenders - owners whose dog had worried or caused damage to livestock before - highlighting the importance of reporting all such incidents to the police.
Warning posters are being displayed in the area where the attacks took place, and police are urging anyone with information about the attack to ring North Yorkshire Police on 101 and quote reference number 12210071036.
There have been several other incidences of sheep worrying in recent weeks.
Last month a woman fled the scene after being challenged by a farm worker when her dog, which was not on a lead, attacked a sheep in Drax, near Selby.
A sheep was also fatally injured in an attack in Great Ayton in early February, and during the same week in Dunnington, near York, a dog owner agreed to pay veterinary bills for the injuries inflicted by his husky in an attack.
Earlier this week a passer-by filmed a Staffordshire bull terrier chasing a flock of sheep in Nether Haugh, near Rotherham, causing them significant distress and leaving two with severe injuries.
Farm owner Nick Grayson said: "The dog has ripped one of the sheep's chin and left her with a hole which means whatever she eats, it just leaks out.
"The other sheep has puncture marks on her face and puncture wounds to her leg.
"All 60 of them were chased around for a good 15 minutes and that would have been really stressful for them.
"We are due to start lambing in a couple of weeks and I just don't know what is going to happen because of the stress of the attack.
"There could be dead lambs inside them. It is just a waiting game at the moment."
He also added that dog owners should never assume their pet would be incapable of aggressive behaviour towards livestock and urged them to keep them on a lead at all times when near farms.