Phin, who was extremely fleet of foot, chased after a million squirrels, often coming close. We used to ask: what would he do if he ever caught one? I have no idea.
Sassy did have one unfortunate encounter last fall. She was leaping through the air, going as fast as she could and bumped into a tree. She laid there like she was badly hurt for long enough for me to have to pick her up and carry her a bit, but then she shook it off and she was fine.
By the way, there are two distinct populations of squirrels around here. Around the house we have the conventional gray squirrels. By the barn and down the trail from there we have black squirrels, and they have been there through at least several generations.