I was diagnosed with selective mutism when I was 4. I'm now 16 and recovered. With therapy and lots of support and trust I was able to make friends, open up to others, and by middle school, talk to new people freely. Some people keep their SM until adulthood and don't recover until later in life, but I was lucky to be able to become verbal while still in school. I had tons of really bad anxiety during middle school, but I also got over that as well. I still get anxiety when it comes to presenting things in front of a classroom setting, but that I can probably work around, too. I'm a mentally ill teenager, so of course I still have things going on, but as far as SM goes, I'm verbal, have plenty of friends, and I'm overall successful. Your daughter can definitely recover as soon as I did if you give her the proper treatment, love, and care.
In my early years of elementary school I saw a therapist who slowly helped me open up to others and have an easier time talking to people. I recommend getting a therapist for your daughter to work with her at school and help her with social anxiety, as well as getting disability accommodations for her. I don't know how it might work in other countries, but in the US you can get her a 504 Plan. That way if she doesn't want to present something in front of the class or answer a questions or anything of the sorts, she cannot be forced to (because no matter how forced, SM will prevent her from speaking no matter what).
Also, I think you should be familiar with the symptoms of SM, even more obscure ones, so you can identify them if you see them. Also, like clairekitty said,
really don't try to force her to do or say things that obviously make her uncomfortable. When I was little and would want a free cookie from the store, my dad would get upset with me if I couldn't say thank you to the employee and sometimes not let me at all, which reduced me to tears. And yeah, don't make a big deal out of her talking or making progress, either. Treat it as if it's normal and maybe give her a reward.
Be flexible and understanding. Don't make her feel like her disability is an inconvenience for you.
If she's like me, she might be very self conscious. People with SM are afraid of being put on the spot, being ridiculed, being laughed at, and have general anxiety issues. I would start crying at the dinner table just because my family would laugh at me for saying something "cute" (as anyone might) and I felt like I was being ridiculed.
Having SM as a child... there were things that I still remember and am affected by even to this day. It goes away, but things still linger. I'm awful with small talk (absolutely hate it) and tend to talk less when I'm anxious or in a bad mood. Still, I'm always improving, or at least I'd like to think.
Sorry that this was kind of all over the place! I wish your family the best of luck, and please don't hesitate to send me a private message if you have questions.