I am a big dog lover, and to answer your question, dogs, like human beings, can face their own set of physical and cognitive challenges. It's not necessarily a matter of being 'retarded,' which isn't a term we use anymore due to its pejorative connotations, but more about creating an environment where dogs with special needs can thrive.
Dogs have been known to exhibit signs of what we humans would classify as mental disabilities, including canine cognitive dysfunction (CCD), which is akin to Alzheimer's in people. Showering these dogs with love, patience, appropriate care and medical attention can improve their quality of life.