Manwe's ordinance at the end of the First Age was that all those who have even a drop of Mortal blood will be Mortal by nature, but to Earendil and Elwing, and their descendents, he gave a choice: to be counted among Elves or be counted among Men.
This is covered in HoME 5 (The Lost Road) in the section dealing with the end of the 1937 Quenta Silmarillion, and the text was never subsequently rejected or revised by JRR Tolkien, although Christopher Tolkien editorially removed it from the published Silmarillion for unknown reasons.
The full reading is:
Now all those who have the blood of mortal Men, in whatever part, great or small, are mortal, unless other doom be granted to them...
Furthermore, in many places Tolkien explicitly states that - aside from issues of Mortality and Immortality - Elves and Men are actually the same species. Here's an extract from the Athrabeth Finrod ah Andreth (published in HoME 10: Morgoth's Ring):
The existence of Elves: that is of a race of beings closely akin to Men, so closely indeed that they must be regarded as physically (or biologically) simply branches of the same race.
The conclusion is that Half-elves in Tolkien aren't actually a separate species at all. Instead, the offspring of Men and Elves are Men (hence the Princes of Dol Amroth), with the exception of the descendents of Earendil and Elwing, who - once they make their choice - become the type they choose to be.
A slight complication is that choosing to be a Man is irrevocable for both you and your descendents (this is nowhere stated, but otherwise the Heirs of Elros would have had a similar choice, whereas they didn't) but choosing to be an Elf confers the ability to choose on your descendents (otherwise Arwen, Elladan and Elrohir would have already had their choice made for them).
So when Elrond choose immortality at the end of the First Age/start of the Second Age, he actually became an Elf, and likewise Arwen became a Man.
So yes, she died.