I believe that the Ring-Bearers and Gimli did in fact die, some time after reaching Valinor. Bilbo, Frodo and Sam went there for healing after their ordeal with the Ring, a grace that was granted to them when Arwen chose mortality. "For I am the daughter of Elrond. I will not go with him when he goes to the Havens. For mine is the choice of Luthien, and as she, so have I chosen, both the sweet and the bitter. But in my stead, you shall go, Ring-Bearer, when the time comes and if you so choose." The Return of the King, Many Partings.
"For I am the daughter of Elrond. I will not go with him when he goes to the Havens. For mine is the choice of Lúthien, and as she, so have I chosen, both the sweet and the bitter. But in my stead, you shall go, Ring-Bearer, when the time comes and if you so choose."
The Return of the King, Many Partings.
As for Gimli, he went with Legolas after the passing of King Elessar, because he desired to see the Lady Galadriel again, and it is said that she obtained this grace for him. But it does not say that he was granted immortality, only that he was allowed to go into the West.
