No, it is not mine. I am not attached or identified with anything. It's just that your idea of what that looks like is inaccurate.

I think you are mistaking "taking care of something" as ego or self-image. If I want to take care of my arm, that is not self-image. That is just something I want to do which does not involve a self, because my arm is not imaginary. Your arm is not a self, and it is not "mine", it does not need either of those things to exist and be cared for, because both of those are literally just made of imagination. Again, you do not need to imagine something that does not exist to survive. Your imagination still operates after enlightenment, that is why I can worry about my arm, however, the imagination does not waste energy imagining a separate self. Taking care of something is not attachment or identity, you don't need either of those to take care of something. If I started worrying about future or past events related to my arm, that would be self-image. Taking care of it in the present moment is not self-image though.

Your idea of no-self is not actually no-self. Your idea of no-self is that you can't feel pain, or that you can't take care of things. This is not what "no-self" really is, and you can see this by looking at anyone else you consider enlightened, because they are all functional and do show care towards things.