Picking skin until it bleeds

[NLima]

My 3 year old daughter got into this "habit" of scratching /picking her face until it bleeds. Then it gets a scab and she keeps taking it off constantly, so it never heals. It's been 3 months!
It started on the chin, the gp prescribed fucibet and it actually worked but from there she started scratching her nose and now ears. So at the moment she has these 3 scabs that she keeps picking on.
I don't know what to do anymore, I tried plasters, creams, making her aware of it, ignoring... Nothing really works.
Anyone with the same problem? I am so worried it will turn into a "forever" thing! I read online that it can be related to anxiety, but she does it straight away after waking up in bed, how can she be anxious in bed?
Also, it actually started at the same time she was toilet trained, so I don't know if it has anything to do with it.
She says it's itchy, but I don't know if it's true.
Thank you.

Honestly, I'm not sure on this one but she could be anxious, even on waking. My DH has anxiety and you'd be surprised at the times of day he's anxious and what he gets anxious about.

Is she in any kind of Childcare yet and what do they think?

Have you spoken to your HV too?

@CadyEastman thanks so much for your reply.
Oh, I honestly have no idea what she could be anxious about, nothing in her life changed at all, apart from toilet training.
She's not in daycare yet, but will start this year. And I hope until then this problem will go away as I am scared she will get worse with that change.
I left an email to the HV but no reply yet, so I am calling on Monday. Also, she has a gp appointment next week. Fingers crossed