Rented an apartment to a person, and lease has expired and not being renewed, tenant still living in the apartment without a lease and has not paid for 3 months, allegedly fell and broke an arm and now is suing me.
Do I have a serious Lawsuit in my hands now?

The tenancy at suffrage and the fall are two separate things.

Why are they still in possession of the apartment?

Was the fall due to a broken staircase or something similar?

Mu questions too - why did you allow this tenant in the apt for those three months without payment? Why did you not give notice to evict? How did this person fall? Can this person prove that the fall was due to something faulty on your part? The tenant is basically a squatter. I suggest you get a lawyer and asap.

I would evict them immediately unless you know for a fact that his broken arm was caused by something in the building. Then get a lawyer and countersue for the rent up till the end of the lease and see what happens in court.

It depends. What is the tenant alleging caused the fall? Did the tenant just slip and fall on your premises with no apparent cause? Was there a defect in the apartment, such as buckled foam under a carpet that you knew about but the tenant didn't know about until the fall? Was the defect caused by the tenant? Was there a defect at all? Did it happen inside or outside the apartment?

A tenant is a mere licensee. What this means is that he or she has no valid claim unless he or she can prove willful or wanton injury. Willful or wanton basically means you wanted to hurt him or her or otherwise let him or her encounter a danger and you didn't care if they got hurt. You generally have no duty to inspect the property to uncover any otherwise hidden defects, but you do have a duty to disclose and repair hidden defects you actually know about. You also can't create hazards designed to hurt the person on your property, obviously.

Call your insurance company, presuming you have premises/property insurance, and tell them immediately of the suit and fax a copy of the suit. Don't let 30 days go by because you have to answer within 30 days max under most state laws. If you have insurance, your insurance company will hire a lawyer for you who will interview everyone, get the facts, then go to court with you. I cut my eye teeth defending this kind of case, and I never lost one. Defense verdicts every time. Now of course, whether you win or lose will depend on the facts of the case.

If you don't have insurance, call a lawyer.