Your dog is likely a terrier breed. Terrier breeds were created to go after prey, so their prey drive is not always controllable.
I can only recommend to run a line across the yard and tie your dog to it so it has a lot of room to roam the yard without being able to get out.
Neighbors in base housing had a chow. He had killed three cats at their previous base and my wife watched him kill one of our other neighbors’ cats. Very traumatic for her, she’s a cat lover (and was 8 months pregnant, flew down the steps to try to save the cat).
My bull terrier is defective. Her best buddy is one of our cats...