Is the Dalai Lama not permitted/unable to attain enlightenment?

Because it's passed down through reincarnation, and the act of achieving enlightenment essentially allows you to exist outside Samara, wouldn't the first Dalai Lama to become a Buddha also become the last that physically exists?

So is the title of Dalai Lama inclusive of the fact that they are an individual who is destined to perpetually experience death and rebirth?

I am aware that the Dalai Lama isn't recognised as a spiritual leader by most branches of Buddhism but I was curious how this spiritual hurdle is handled in the Tibetan version.

Thank you

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