TITLE: Witt-vector vectors
QUESTION [16 upvotes]: I've never really made my way in any detail through the Witt-vector construction.  I did read all the articles that a quick Google and MSN search turned up, and none seemed to address it, but I could just be unfamiliar with the language; so please pardon me if this is a question with a well known answer.
If it makes a difference whether I'm asking my question about the 'big' Witt vectors or the $p$-typical ones, then please interpret it in the way that makes it more interesting.
With that prelude, the question itself is simple:  is there a functor $\text{$A$-Mod} \to \text{$W(A)$-Mod}$, either just for specific algebras $A$ (I have in mind the $p$-typical Witt vectors of a finite field of characteristic $p$), or, possibly, for all commutative algebras?  In my motivating case, one can do something awkward like pick a basis for an $A$-module (i.e., a vector space) and simply construct the free $W(A)$-module on that set; but, aside from the ugliness of this approach, it's not even clear to me that it's functorial.
EDIT:  As both nfdc23 and QiaochuYuan point out, I have additional assumptions in mind; namely, in both cases, that free, finite-rank $A$-modules are taken to non-$0$, free, finite-rank $W(A)$-modules, for nfdc's post that the identity morphism is taken to the identity morphism, and for Qiaochu's post that not every morphism is sent to $0$.

REPLY [9 votes]: Let me restrict my attention to the only case I understand, which is $A = \mathbb{F}_p, W(A) = \mathbb{Z}_p$ (so here I mean $p$-typical Witt vectors). You don't provide any conditions you want your functor to satisfy, but I presume you don't just want e.g. the zero functor, so here is a very simple no-go for getting extra properties: no such functor can both

be additive, and
send finite free modules to finite free modules.

This is for the very simple reason that $\mathbb{F}_p$ (the endomorphism ring of a finite free $\mathbb{F}_p$-module of rank $1$) admits no ring homomorphism to $M_n(\mathbb{Z}_p)$ for any $n$, since the latter is torsion-free. 
More generally this argument shows that the image of any additive functor from $\mathbb{F}_p$-modules to $\mathbb{Z}_p$-modules necessarily lands in modules whose underlying abelian groups are $p$-torsion. Of course there is a natural additive functor given by restriction of scalars along the quotient map $\mathbb{Z}_p \to \mathbb{F}_p$ with this property.