Is it possible for i.e. an Arhat to become unenlightened after he reached the state of Nirvana?
What if an enlightened being gets a severe brain injury by some cause leaving him with amnesia. Could it affect one's enlightenment in any way?
No! Once you attain Nibbana, the defilements are annihilated without the possibility of return. Just like uprooting a tree instead of chopping it down or trimming it. So it can never grow back from roots. Nibbana does not depend on the condition of the body or any other thing. It is uncaused, unborn and permanent. Venerable Moggalana was beaten up by thugs until his bones were pulverized. Still his mind was at perfect peace. One Arahath thero in Sri Lanka was thrown into a cauldron of boiling oil by the order of a King. That even caused a Tsunami in sri Lanka. But he was still at perfect peace when his life ended.
Take a Sothapanna person as an example. He can die 6 times before attaining Arhathship. If death itself is not going to affect his attainment, how can brain damage affect it?
No, an Arahant cannot become unenlightened. Does that mean that an Arahant will automatically be free from illnesses that affect the mind? I don't think so. They will always be free from the three unwholesome roots of greed, hatred, and delusion, but I think it is a serious mistake to translate that into physical claims about Arahants.
Interestingly enough questions similar to this were a point of controversy between the Theravada school and other early schools. Some of the other schools claimed for example that an Arahant was physically incapable of having sexual arousal, and so could not have a nocturnal emission. The Theravada school rejected this, saying that such a thing is purely physical.
Arahants can still experience the purely physical aspects of things without having unwholesome mind states. If you inject an Arahant with adrenaline, their heart rate would increase and they would undergo all of the reactions of a person with fear, but the difference would be that there would be no aversion in the experience, even if they might physically present the signs of it.
Similarly, I think that an Arahant could have what are usually refereed to as mental illnesses, although I think that term is somewhat misleading in the case of an Arahant. Perhaps the term neurological illness would be more appropriate. If they get Alzheimer's disease, then they would undergo all the memory problems and such, but throughout all of this there would be no greed, hatred, or delusion in the mind.
Someone mentioned PTSD. I think an Arahant could have PTSD as well, having the flashback experiences and the physical arousal that goes along with it, but again, without greed, hatred, or delusion.
That's just my take on it.
It's possibly worth adding to this that although the Theravadan school states that once a being is an Arhat they can't regress this wasn't a universal view within the early Buddhist schools. Both the Pudgalavada and the Sarvastivada schools did maintain that an Arhat could indeed regress from that state. So for those schools it would have been possible to become unenlightened. This was the fifth point in some versions of the five points of Mahadeva which are a number of debating points about the qualities and attributes of an Arahat. However since these schools have since died out this concept of unenlightenment would be of historical interest only.
Just as a reference note this is from An Introduction to Buddism by Peter Harvey (page 97 in the second edition).
If I am to add to the existing answers, it is like a singularity where all conventional laws break down. So what happens beyond on in this state cannot be communicated through conventional wisdom or language.
Nirvana is the experience beyond the singularity. Once you experience it you have experienced it. You cannot unexperience it. Say you have gone and see Rome. This is a fact. Is it possible to turn back time so you have not seen Rome. Not possible. Likewise once you have seen what is beyond the singularity you cannot be made to be brought to the state where you have not seen it.
Also you experience Nirvana when you are in Pala Samapathi. When you come out of it you don't experience it but the fact you have seen it will not change. Since you know how you got there you can get to it when you want. This experience has a lasting impression / imprint on you. So your ways will change once you come out of the Samapathi.
To further elaborate. The lasting impression is like once your where in Rome had a life changing experience. This sticks.