Yes, every mortal who goes to Aman (the Undying Lands) will eventually die. The Undying Lands were likely called that because immortals dwelled in them, not because they granted immortality.
Other important arguments against the immortality of the mortals who sailed to Aman can be found in a letter:
As for Frodo or other mortals, they could only dwell in Aman for a limited time – whether brief or long. The Valar had neither the power nor the right to confer 'immortality' upon them. Their sojourn was a 'purgatory', but one of peace and healing and they would eventually pass away (die at their own desire and of free will) to destinations of which the Elves knew nothing."
And in a passage from The Akallabêth:
The Eldar reported these words to the Valar, and Manwë was grieved, seeing a cloud gather on the noontide of Númenor. And he sent messengers to the Dúnedain, who spoke earnestly to the King, and to all who would listen, concerning the fate and fashion of the world.
‘The Doom of the World,’ they said, ‘One alone can change who made it. And were you so to voyage that escaping all deceits and snares you came indeed to Aman, the Blessed Realm, little would it profit you. For it is not the land of Manwë that makes its people deathless, but the Deathless that dwell therein have hallowed the land; and there you would but wither and grow weary the sooner, as moths in a light too strong and steadfast.’
—J.R.R. Tolkien, Christopher Tolkien (ed.), The Silmarillion, "Akallabêth: The Downfall of Númenor"
