Pumping Lemma for regular language
The pumping lemma says that every (long-enough) string in a regular language can be divided in some way such that it can be pumped (subject to the other stated conditions).

Note that the statement is that every string has at least one way of dividing it up. Not every division of a string necessarily works. As you have shown, some don't work, however there is a different division (any one where $m$ is even) that does work. So the language does satisfy the pumping lemma.