Richard Sharvin, who was believed to be in his 70s, died after the accident that involved a tractor took place close to Strangford village on Friday.
Police attended the scene along with officials from the Health and Safety Executive (HSE) who will lead the investigation.
South Down MP Margaret Ritchie said people in the area were in "deep shock" after the death of the father of two.
"I want to offer my deepest sympathies and condolences to Richard's widow and to his daughters," the MP said.
"The local community will stand shoulder to shoulder with them in their tragic loss."
Ms Ritchie added: "Richard was a very upstanding member of the community - loved his farm, loved farming enterprise and he was very respected among his peers."
The HSE said it was aware of the incident "and are investigating".
An elderly farmer has died in an accident on a farm in County Down.