There's nothing under Connecticut landlord-tenant law that would prevent a tenant from entering into a lease that automatically renews absent notice of intent to vacate prior to the renewal term.
You state that the lease was for one year "and she stayed 2.3 months". I'm assuming you mean she stayed 2.3 months into her second year.
Since the tenant did not vacate and remained after the renewal date before giving notice more than one month later, the tenant was obligated under the lease for the full year term. The tenant could not therefore terminate the lease early simply by giving notice.
Unless a new tenant were found, the original tenant is obligated for the rent for the entire year period. However, you have the responsibility to mitigate damages, which means using reasonable efforts (e.g. newspaper ad) to try to find replacement tenants.
Since the tenant was obligated for the year, you may retain the security deposit to compensate for unpaid rent. You still must send the tenant a letter indicating the deductions you made from the security deposit and include any balance left over (not likely in your situation). That must be done within thirty days of the tenant vacating, if the tenant gave written notification of their forwarding address. If not, then you have fifteen days from the date you receive such notice. However, regardless of whether you receive such notice, you should respond when you can in order to wrap things up.
Please let me know, if you have follow-up questions.
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Attorney
Attorney and small business owner with 8 years experience in the general practice of law.