Hi
My general rule of thumb is not to take a pupil until their 6th birthday. Generally I only take them on at this young age if they have a parent or someone who will supervise their practice and provide ongoing support. If not I advise that they wait a year or so.I have been teaching for over 20 years and formulated this rule gradually. There are lots of musical things that you can do which will help your child succeed brilliantly when she does start formal piano lessons. Any rhythm games, singing games, clapping, listening to all sorts of music and dancing around et etc. If she wants to explore at the piano, great. If she is musical and bright there is no harm in waiting a couple of years. Of course it is not insulting for you to want her to have lessons with your teacher but I would advise waiting. 6 year olds that I have taught who were more than ready to learn, just take off and fly. On the other hand several children I have 'inherited' who started lessons at 4 had all developed bad habits and really were no more advanced than children who started at 7 or even 8.
However if you are very keen to get her started make sure you find an experienced teacher. The early stages are vital and at 4 habits, good and bad, will become very ingrained. Good luck