No kill at least in Australia means you can’t put animals to sleep due to over capacity, time frame etc. The only time they can be put down is when they’ve attacked or are showing high signs of aggression and the behaviour assessor finds they aren’t suitable for readoption.

At that point, it becomes a duty of care to put the animal down as it’s cruel to keep it in a kennel for the rest of its life and it can’t be trusted as a family pet.