Hi there, yes that small piece is dead and may eventually fall off. As long as it does not spread up the tail it will be fine . You can get some betadine from a drug store , dilute with half water and soak it once a day. Use a very small jar or cup to put the mixture in and just hold him with his tail dangling in it. Or you can dab it on 2-3 X a day and put neosporin on at night.