Dear Monica,
After three months how did you assess her performance? Do you have well-defined criteria and it is just manager's judgement? By the way, have you spoken to her? Why is she refusing to become permanent employee?
What are the terms of employment as per appointment letter issued to her? Though your post is small, one gets a feeling that she may not be comfortable in continuing employment in your company. She has been looking for a change and should need arise, she would like to quit the company. Possibly up in her sleeves she has a job already and she could be waiting for receipt of offer letter.
While quitting she would like to take benefit of shorter notice period rather than longer one when she becomes confirmed employee. The probable reasons taking this decision is her lesser incompatibility to your organisation's culture, uneasiness with working with her manager, less cooperation by the colleagues and above all job with better salary offer.
Sit with her and try to understand the reasons of her decision. You may convince her to stay but do not cajole her. If you find the reasons invalid, then you may start searching her replacement. In the meanwhile, issue her letter for the extension of the probation period. Sooner you get proper replacement, you may terminate her services.
It is important for the employees to give their best. What is the use of keeping those employees who wish to work half-heartedly?
Once you find out the reasons, we request you to come back and update us on the latest development.
Thanks,
Dinesh V Divekar