A nice wee book
https://www.amazon.com/Magic-Years-Understanding-Handling-Childhood/dp/0684825503

BTW this is not uncommon at all at this age.
Look to possible factors like;
a cold toilet seat
fear of closed door
pain by passing stool / constipation
enough fiber and water intake
fear of letting go of something you produce
fear of the flush (many kids have also a fear of water going down the drain/fantasy they may go down the drain too, fear for vacuum cleaners/magic thinking)

Sounds fear related and I would take all pressure off and have an relaxed approach.
I’m sure the problem is solved by now !

Oh that is funny as it now reminds me of my son when he was like that and boy could he hold it in . I would not have expected that from him and he was really fussy in that regards when it came to toilets. But there is going to be a time when he ad to go and I took him to the ladies as would not trust a young child to go to the gents especially on his own . Maybe bring some wet wipes and tell her you will wipe the seat for her to make it clean ? .Good luck

Make sure she drinks lots and lots, that will make it harder for her to hold on. My daughter went through a phase of not wanting to use the loo. Ended being because of pain caused by an anal fissure after a bout of constipation. This doesn’t seem to be your problem though

My friends daughter experienced this. It started when she put her daughter into daycare. Normally the worker aides there make the children all go one after the other. It was just a line of toilets in the open so no privacy for any of the kids. if your daycare is like this i am sure thats were your problem lies

that’s such a long time to hold on, holding on can cause urine infections :(, maybe a reward chart, im going through toileting issues with my 3 year old will wee on the toilet but wont sit to do number 2s does it outside in his room or sometimes even right in front of the toilet, i would suggest being patient , may be going in with her shutting the door, the childcare centre may be able to work with you using the same techniques you are for consistency. Hope things start to improve for you tough one with no real answer x

My granddaughter had a problem with doing No. 2’s on the toilet & insisted she have a nappy on to do it. She had no problem weeing on the toilet, & with all our promised treats & bribes nothing seemed to work until her Mum & Dad said they are not buying anymore nappies. It worked!

Ah, that’s really tough. If it was just at home I’d maybe have a radio or music playing in the bathroom in case it’s because she doesn’t want to be heard? My son (now 15) has always hated going to toilet in public and it has caused some issues with constipation because he holds on all day at school, so I hope you can get this sorted out.

One of mine would refuse to do number twos anywhere but home , it was shocking so he would get to the stage where he couldn’t hold on any more and do it in his pants , we took him camping for 5 days what a nightmare that was he would hang on then he would get to the toilet we wouldn’t go and say I don’t need to go now in the end I had to just make him sit there as we didn’t have an option especially camping , I just made a point to visit the toilets more when we were out and about and eventually he out grew it , can be frustrating , good luck