Part: Noun.

Well, it seems that my fancy about part being a determinative was just that. In an e-mail, John Payne pointed me to the discussion of half in the CGEL, where he and Rodney Huddleston argue that half and other fractions are nouns. As evidence, they remind us that they can be plural and can take a determiner, two key properties of nouns. Indeed, the same thing is true of part.

1. It's part table and part desk.
1b. He's one part father and two parts geek.
2. Part of the money is gone.
2b. This part of the picture is missing.

Geoffrey Pullum, in another mail writes,

(Part) participates in a number of idiomatic constructions with anarthrous nominals: compare "part man, part animal" with "step by step" or "bog Irish" or "cap in hand" or "hand on heart" or "foot in mouth" or "weather permitting" (etc. etc.): there are hundreds of special phrases in which nouns that normally need an article don't have them.

Finally, Rodney Huddleston takes it further, arguing that, despite what some dictionaries (e.g., the Longman Dictionary of Contemporary English) say, part is still a noun, even when it modifies an adjective or verb as in:

3. The object was part hidden by the grass.

Note that NPs can function as adjunct in clause structure, as in "I saw her THIS MORNING", and we argue that this holds for single word NPs too, as in "I saw her YESTERDAY". There's no more reason for assigning yesterday to different CATEGORIES in this and "Yesterday was completely wasted" than there is for assigning this morning to different categories in the corresponding examples where it replaces yesterday.

So, it looks like part is just a noun, albeit one that can appear in quite a range of functions, except, I suppose, when it's a verb. Too bad so many dictionaries have only got it part right.