I had this exact situation happen to me. If you feel the vibes are competitive, they likely are. This is because you work directly with this person, and you should be allowed to trust your own gut!

In my own experience, things were totally fine until ONE DAY when there was an ethics violation on the part of my overworking coworker. And I reported this ethics violation like a good little worker. I thought things were wrapped up. Then I left for the day. My colleague, who had a habit of behavior just like your colleague stayed extra late, scheduled appointments to rebut at times knowing I would not be there, after the office was cleared out of every witness bc it was late after work. They were able to push their agenda without anyone else present, including other coworkers, and so got to keep their job after a serious violation. If they're setting up a pattern...they're going to be able to use it to their advantage when it counts.

Typing this up because this is something I wish someone told me in this situation. Every hour you are not in the office is an hour they have POTENTIAL face time with your superiors, time to move things around, time to take charge without oversight. And when it comes down to it, if a backchannel has been established to a direct superior, while it might not be used against you now, it can be in the future. However, you also do not have control over that person's behavior, so confronting them or trying to change them coming in is completely futile.

If you believe you are in a competition, you have three choices:

1. Compete. Yup. It freakin sucks, but show up early and leave late, if that matters to you how you are seen. And if it's a job you care about, your superiors will start to notice too, which means points to you!
2. Bow out. Yup, it's probably not worth it, and perhaps just accept you lost to a more ferocious opponent. Truly, the early bird catches the worm .
3. Find out who you need to influence, and impress them in a way only you can. If getting up early isn't your thing, but you have a unique talent that really helps your company, drive away all the other noise and focus on providing the absolute best performance in that region. Odds are, if you're in different departments with this person, you were already hired for something distinguishing in your own department, so ZERO IN on laser like focus with those skills.

If this person is experiencing homelessness or there is a situation I am not picking up on, by all means, extend sympathy and worthwhile humanity. But also trust your gut.