No, the self completely vanishes because it is seen as a non-existent entity. That is what enlightenment is. 

You can still care for family, just not your imagination of them and how it relates to you. It's just that simple. Attachment is just clinging to your imagination of things. Attachment is fear of losing things in the future, but this type of fear cannot exist without an imagined self. There are no degrees of attachment if you cannot imagine yourself, there is just "what can I do right now?" Because I am someone who likes to care about things. Nothing more to it.