A lot of kids have stuff about clothes that their parents can't quite get, sensory kids, for example, just hate stiff clothing, especially things like jeans. It sounds like she either does not like her bare arms to be touched, or is self-conscious about how they look. (When I was young I used to think my upper arms were too skinny, and would refuse to wear shirts with a certain length of sleeve because they cut off right where the skinny part was and showed it off.) I guess the other possibility is that she thinks she looks fat (has anyone teased her about that?) and that the shorter-sleeved shirts seem to her to show off her tummy, but you don't say she is complaining that her waist shows, but her arms. My son, who is 8, says maybe she doesn't want to be exposed, something he feels about shorts. He doesn't like the temperature change from wearing long pants and doesn't like the way many shorts feel heavier than his (lightweight) long pants. Point is, even if you don't know her reason, she has a reason and it matters to her. Your best bet is to work within the set of boundaries created by her reason.
My suggestion is to get her some lightweight long-sleeved shirts. A sweater really *is* too hot in summer, unless it's a mesh sweater, but there are uv-protection shirts and just light-colored or white long-sleeved button-front shirts (like a man's dress shirt). Nothing fancy is needed, I'm sure you could get either a girl's or boy's white long-sleeved shirt and she would be about as comfortable in that as she would in short sleeves, and it would not stand out, even in summer. (This kind of thing is easily available at kids' resale shops for pennies on the dollar, if you are worried about having to refurbish her wardrobe for what seems to be a whim.) If you need to explain to family why she is wearing a long-sleeved shirt in summer, just say it's for sun protection, so she won't feel embarrassed.
She'll probably grow out of this dislike of short sleeves, as her fashion sense takes over and she sees cute stuff with sleeves that don't reach her wrist. If not, or if it expands to other stuff to the point where it is a constant problem to get her dressed, talk to a counselor. But a little calm management now should get her a wardrobe that takes into account her preferences even if you can't quite tell what they are. :)
Does she have other quirks that seem unreasonable or stand out like this?
Sounds like sensory issues possibly to me, too, and anxiety.
Some kids feel much less anxious in clothes that "hug" them. Have you tried her in long sleeve underarmour shirts that are lightly snug all over?