I agree with this. Unless you have moved home/school and separation anxiety triggered (which should then get better after the child feels more secure/resigned to it), then if she doesn't do this anywhere else, it is an issue at school. The fact that she clings to you is that she is trying to communicate her fear to you, as you are her Protector. I should say that when I say a 'problem' at school, I just mean something that has at one point triggered fear in her and her self-defence mechanism, and taken hold. It could be something entirely innocent really, like seeing a 'scary' poster (or one that scared her but nobody else).
Sadly one of the big things about a problem at school is that kids often cannot or will not articulate it. My son had a huge problem with something that did actually start in Year 1 and got way worse in Year 2, and he didn't tell me about what it was until he was about 11 years old.
In his case, he'd seen a short video clip in the after school club about meteors hitting the earth (I think maybe a trailer for a show), and it had terrified him. From this came a massive dreadful school anxiety (made worse by a couple of mis-managed age-inappropriate other situations by school).
Your DD could be having an issue with adnother child, or she could be finding some of the practical ways at school difficult (toilets, eating, sitting still, group work, playtime, etc.). She could really have seen something that scared her (I saw bad boys breaking eggs in a bird's nest and I would never go beyond a certain point in the playground after that), or she could even have some anxiety about something happening to you while she's not with you.
Either way, with hindsight I would keep talking to her, asking her about every aspect of her school day, but in a subtle way. Like 'oh is play time nice? Who do you play with? Where do you go? What do you do if it's raining? Has anybody ever got hurt? / etc.' (she may have seen something happen to another kid and got scared). I might try something along the lines of is there anything at all ever scary at school.
The absolute horror at being left at school is a self-preservation move. Very strong. It is true that if you force her, this will stay with her. So it would be way better to somehow unravel it. I hope you do x