Officers have also been granted additional time to question the suspect, who police have revealed is a 33-year-old man from the Leeds area.
Officers will remain at the scene for several days after a woman's body was discovered in a house on High Street in Thornton-le-Dale, shortly before midday on Wednesday January 5.
The man was arrested shortly afterwards and remains in custody.
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The woman has not been named, and floral tributes have been left outside the property.
It is custom for someone who has been arrested to be held for up 24 hours before they must be charged with a crime. In cases of the most serious crimes, this can be extended for up to 36 or 96 hours.
A spokesperson for North Yorkshire Police said: "Further updates will be provided when possible and we continue to ask anyone with information which could assist the investigation to make contact by calling 101 and quoting reference number: 12220002314."
In an earlier statement, North Yorkshire Police Detective Superintendent Wayne Fox said: "Enquiries are continuing in our efforts to establish the circumstances which led to the death of a local woman.
"I want to reassure the local community that, whilst this is a tragic event, North Yorkshire Police are satisfied that we are currently dealing with an isolated incident.
"Our thoughts and sincere condolences go out to the family of the woman who sadly lost her life and they are currently being supported by specially trained officers.
"You will see an increased policing presence in the local area. Please do feel free to speak to our officers if you have any concerns."