Perkins made the case that up to 100 cast and crew would be working on the shoot and would inject a considerable amount of cash into local businesses including hotels, rentals and restaurants.
But Shelter Island councilman Jim Colligan had a daughter in the film industry and was concerned it would create a "whole mess".
However, not all of Shelter Island's 2500 residents are pleased with the decision.
A local resident told the Post: "The councillors made this big deal about filming, when it would have given the locals a lot of money in food and rentals. People are annoyed."
The Undoing is an upcoming mini-series based on the 2014 novel You Should Have Known by Jean Hanff Korelitz.
Kidman also serves on the project as executive producer.