No, The phrase "Undying Lands" is not a place of immortality bestowed. It was a realm inhabited by Ainur (spiritual beings) and Elves whom are already immortal. The "Undying Lands" do not grant immortality, the Valar do not have the power; OR the authority to bestow immortality upon mortals (though they did "bless" the Men loyal to them during the War of the Wrath, the "Numenoreans", with extended lifespans).
The Undying Lands are simply named for the immortal inhabitants. The Ringbearers travel there to receive healing for the wounds (spiritual or physical) incurred as a result of ownership of the One Ring; the effects of the Ring is ubiquitous (albeit differing in time) to all beings immortal or otherwise. They would still die, but the hope was that they could live the rest of the time without Suffering or if they could be healed; allowed to return. The elves that bore rings of power also went there after the power of their rings was "spent" thus their domains would deteriorate.
In Tolkiens letters he describes...
Frodo was sent or allowed to pass over Sea to heal him - if that could be done, before he died. He would have eventually to 'pass away': no mortal could, or can, abide for ever on earth, or within Time.
Also stuff in Valinor happens very... very slowly since all its usual inhabitants are immortal there's little need to rush tasks, and 100 years would seem like just a mere blink of an eye, so really its not a nice place for a mortal person to reside; less they're left to rest.