Seeking asylum? Sorry we are racist...

[Jurion]

And tales and rumours arose along the shores of the sea concerning mariners and men forlorn upon the water who, by some fate or grace or favour of the Valar, had entered in upon the Straight Way and seen the face of the world sink below them, and so had come to the lamplit quays of Avallone, or verily to the last beaches on the margin of Aman, and there had looked upon the White Mountain, dreadful and beautiful, before they died.

Silmarillion, Akallabeth

This suggest that people could reach Aman itself, not just Tol Eressea. And Frodo, Bilbo, Sam and Gimli died, they were only healed for a short time.