Yes, absolutely report it. I agree with everything Kim said. It is totally unprofessional that this caregiver would talk about your issues to other staff and air her dirty laundry to them. It is utterly unacceptable if she is hurting your child. Report to the director, in words and by email. I would even be tempted to report her to authorities.
There should be a grevience policy and procedure in place. Don't go to the authorities before you take the steps in the policy first - which is addressing to director first of all - then if you aren't happy with the outcome you go further. It's only fair to give the centre the opportunity to address it first..