The attacks have all happened in the past fortnight. In the latest, a sheep grazing near Burnt Oak Road was badly mauled and had to be humanely destroyed.
Other local farmers have reported injuries to livestock also thought to be consistent with dog bites.
The dog suspected of carrying out the attacks has been described as a large, tan-coloured animal. When last seen it was wearing a dog coat.
Sergeant Tom Carter, Sussex Police lead for wildlife and rural crime, said: “Livestock worrying has a devastating effect on the sheep farming community and is a nationwide problem.
“Responsible dog ownership is the best prevention. Please keep your dog safe and do not let it roam away from your property. Where appropriate, police will prosecute owners who allow their dogs to attack sheep and under certain circumstances dogs worrying livestock may be shot.”
Anyone able to identify the dog in the High Hurstwood incidents or witnessing similar incidents is asked to report details online or by phoning 101, quoting serial 779 of 20/12.
Alternatively, please visit the Crimestoppers website or contact the independent charity anonymously on 0800 555 111.
Reference Number: SXP51582/2017