How old is your dog? In my experience, six months to a year old is going to be a rough time for training this kind of behavior, because prey drive goes way up and their instincts take over. Different dogs handle it better or worse, but you are asking your dog to go against instinct by not chasing/killing a running chicken. When young, they will often lack the necessary impulse control to stop from chasing, so mistakes can and will happen if you are not acting first or taking extra precautions. They can learn to do it eventually. But that "eventually" can result in a lot of dead chickens, if you are not very careful.
How are your chickens setup? Are they enclosed or free-range? Are you able to keep them physically seperate from your dog?
Ideally, a true livestock guardian dog should have a very low prey drive and natural guard instincts Not all individual dogs are well suited to the task and some breeds are more likely to do well at it. There are several dog breeds that have been bred for this purpose, but even they need to learn what animals are under their care and what not to do as a puppy, it just comes more naturally to them. Herding dogs, like Australian Shepherds, are bred for a different set of skills. They tend to be nippy and love to chase, so that can lead to problems when they are around smaller animals. On the plus, they are people-oriented and highly trainable. They can make excellent general purpose farm dogs and learn to ignore livestock and poultry. They might not understand WHY but they can be trained on HOW to be around prey without chasing. Just keep in mind that they are going to be working against instinct and will need consistant training and enough time to mature and learn. Chasing chickens is VERY rewarding for your dog so you need to be proactive about reducing opportunities to chase.
If your dog is under two years old, I recommend keeping them physically separate from the chickens while you work on desensitiving them to the visual trigger of spotting a chicken. At the same time, work on Recall, Leave It, and Stay, along with whatever other commands you use regularly. You want to be able to call your dog away and be able to break that intense focus with a command. It will take time, so keep the chickens safe. Reward good behavior and minimize opportunities for bad behavior.
Not all dogs are able to overcome their instincts, but most dogs will respond to training and will get there eventually, if you have enough time and patience.
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By the way, I feel your pain. This is my Anatolian Shepherd/St. Bernard mix at ten weeks old, getting introduced to the chickens for the first time. He did great.
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Fast forward to a few months later and I find him on the couch in our living room eating a dead duck. They grow up so fast ....
Fortunately, we got passed that phase and Bear is now living up to his full potential as a livestock guardian, protecting our farm from hawks, coyotes, cougar, and even his own namesake, bears.
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So there is hope.