What is the most basic, most fundamental function that a hero performs, one so integral it can be used as a way of defining the term hero? There are several ways to answer that question, of course, but one would be the act of saving others.
By that definition then, Batman is most certainly a hero, and not merely because of all the fictional women he’s saved from attempted muggings or all the times he’s pulled Robin out of death traps. Batman has saved real people, too, despite the fact that character isn’t himself real in one of the stricter senses of the word.
He saved Dean Trippe, and Trippe’s phenomenal autobiographical comic Something Terrible tells the story of how it happened.