My understanding is that the contract continuation was implicitly continued by both parties- this means that the individual remains on the same terms and conditions as before. I would check this with an employment lawyer though to be safe.

Notice is a tricky thing, he/she could just walk out at any point in time; there is very little come-back from an organisation with regards to forcing the employee to fulfill their obligations. Also, by forcing the individual to remain through their notice period, you are risking their performance in the company and the general impact this can have.

I would recommend that you sit down with the individual and explain that your interpretation of the document is that by remaining in the role and accepting their salary they are implicitly accepting the continuation of the contract. The likelyhood is that they are feeling aggrieved by the length of time the process has taken and messed about.

I would admit that "best practice" had not been followed in this case and that you are reviewing your processes to ensure this never happens agin and thank them for their patience. Ask if their motivation to leave is geared around the absence of a contract for some time and see if there is a path forward.

In future I recommend that in FTC workers' inductions, you set a "contract review date" where they will be met with to discuss their performance and their contract as a whole. This should be on or before their notice date as laid out in their contract. It's either at this point you refuse to renew, effectively giving notice, or furnish them with an extension.