It is believed the three sheep were attacked by dogs in a field at Topcliffe some time overnight on Monday and yesterday.
A spokesperson for North Yorkshire Police said: "Dog attacks on livestock result often in death or severe injury, and have a huge impact on farmers and their livelihoods.
"The message to owners is simple - keep your dog on a lead near livestock, even if they are in a separate field.
"Even the most mild-mannered dog can see sheep, especially lambs, as something to chase and kill."
Anyone who has any information about the attacks in Topcliffe is asked to contact North Yorkshire Police on 101, quoting reference number NYP-02082017-0087.