In the option View> Litters> Puppies you can see all the puppies that have been born in a litter.
If you want to keep a puppy, you can do it and in this way it is added to your list of breeding animals and you can use it to make matings and litters.
Select the puppy and click on the button More options> Add to my list of breeders
It will no longer be available to be reserved or sold
Now, it will appear in the option View > Animals (where all the breeders appear
You can make matings and litters with that animal