To think she comes across as rude?

[applepeartriangle]

I have a 5.5 year old dd.
She’s doing very well at school, but is the youngest in her school year so I think emotionally she may be slightly behind. Academically she’s one of the top abilities according to her teacher.
I just don’t know what to do with her in this situation...
Basically if any of her friends or adults come up to talk to her (eg teaching assistants out of school, or friends parents in the playground, or her school friends we bump into at the park) then she point blank refuses to speak to them.
All of her little friends run up to her in the playground and say hi, and she just slams her eyes to the floor and clams up.
If friends parents ask if she enjoyed a school trip for example then she’ll do exactly the same thing, just go totally mute.
She has lots of friends in school and is being invited to play dates a lot so I don’t think it’s affecting her friendships, but I worry that it comes across as really rude.
We’ve spoken to her about just waving or saying “hello” or even smiling, but she can’t even do this.
She’s very very loud at home, one of the quieter children at school, and totally mute when people approach her out of school time. She’s fine with family.
I just worry at her age that she should be able to manage a simple hello or wave.
Apparently once she’s on a play date then she does talk to her friends parents.
Any advice? I feel like I should be making excuses for her and saying “she’s a bit shy” but not sure if this will make things worse. At the moment she just looks incredibly rude!

I have a dd with selective mutism don't rule it out completely. Dd can often talk at playdates but not when something unexpected happens, sometimes she can speak out in class now (other times not).
The advice I was given was to not make an issue of it. Push the boundaries but not too much. Encourage talking but practice before (eg in a cafe get her to order her own food she can practise while your choosing or pay for something at the till), advice people she sees that this may happen and so to ignore it and reassure she's not being rude.
I also have a ds with asd he would not even talk to his grandma when he saw her unexpectedly but has grown out of that stage now.
I'd talk to teachers re speach therapy they often come into school so it's a safe environment and they did my dds visits with out her even knowing (they observed told ta what they wanted).