The Immortals

[Vishtaspa]

I would not be at all certain that it reproduces any real ancient Persian work.
Is there some other, perhaps Persian, literature/evidence to support this idea?

Here is a excellent Status Quo and general scholar feeling about Cyropaedia:

(Enter keyword "cyropaedia")

http://www.iranica.com/newsite/home/index.isc

Cyropaedia is fiction, and serves Xenophon's purpouses (cited by Paul M) but here (and more in Herodotus) we found iranian or eastern oral influence. In fact, there was no writen ancient iranian literature at all (only Avesta and inscriptions), but were a very rich oral (epic, poetic...) tradition, as later literature like the superb Shah namah demonstrates.(I must say that also some parts of Avesta contain legendary and epic passages) In fact, tales from both Herodotus and Xenophon have been linked with passages in Ferdowsi's Shah namah (Book of Kings, in which mythical, achaemenid and sassanian times are told).
Cyropaedia has many historical mistakes, and sometimes we are sure comparing actual babilonian or old persian inscriptional reports. But this unexactity is what one can expect from oral and traditional sources. Other "sources" of course, were invented or reelaborated by Xenophon serving his purpouses.

regards