We need a small size intelligent being in order hold the puppy carefully and place it in a basket or sth like that.
Robots? Could work, but agents are usually designed to do a specific task. For saving a puppy from a deep norrow well, I don't know if there any robot exists. Anyway, we dont need to call this an "engineering problem". It's rather a biology problem.
Scientifically proven that mature animals have instinct to save the youngsters. I would release down a basket with another dog in it. When it reach the bottom, the mature dog will hold the puppy from the neck, and place it into the basket with itself. So I could bring them all the way up. A small size monkey could work to. We may need a very small training for the saver animal tough, to mitigate the risk of loosing both.