The incident took place between 2-2.30pm on Wednesday April 7, one of the three days of the week when Mulgrave Woods, near Sandsend, are open to the public for recreational use.
The dog was on a path in the woods when it was struck by a black 4x4, and it died later from its injuries.
The driver of the vehicle has been identified and is currently assisting North Yorkshire Police with their enquiries.
The Mulgrave Estate office confirmed that they are aware of the incident and are in contact with police, but did not confirm whether the vehicle involved belonged to the estate.
Mulgrave Woods are also a commercial timber plantation and walkers are warned about forestry traffic using the tracks.
Mulgrave Castle and its surrounding estate are the home of the Marquis and Marchioness of Normanby, whose family have held the land since 1743.
Officers are appealing to anyone who saw the 4x4 prior to the collision to email PC Bethany Thompson at [email protected]