I disagree that switching schools won't help. One of the problems with selective mutism is that peers begin to expect the child not to talk. When the child does speak, it may elicit reactions of surprise from peers. From personal experience, I know that these reactions are very uncomfortable to a person with selective mutism. At a new school, peers won't have these expectations. The Wikipedia page for selective mutism actually does mention switching schools as part of treatment, but it also has this cautionary statement:
However, changing school is worth considering only if the alternative
environment is highly supportive, otherwise a whole new environment
could also be a social shock for the individual and/or deprive them of
any friends or support they have currently.
I do agree with Leopoldo Sparks that early treatment is best and that you should seek professional treatment, but I think you need to consider how the child will react to treatment. From personal experience, I know that there is a chance seeking treatment will cause a backlash: When my parents tried to get me treatment for selective mutism in middle school, I was very offended and angry at them and refused to cooperate. According to The Older Child or Teen with Selective Mutism children of eight or nine years old are considered to be older children and will be able to see through some treatments as attempts to get them to speak, and they might resist treatment.
The article goes on to suggest that older children need to be in control of their own treatment, which I think is true. My advice would be to be very open and supportive with your child when seeking treatment. You need to tell her that you're trying to help her and make sure she agrees to treatment rather than trying to force her. As for determining whether to switch schools, I would ask the child if she thinks that would help and have her make that choice. You said she was bright, so you might even be able to use the opportunity to find a school with a better curriculum.