Is the fixed point property for posets preserved by products? Recall that a partially ordered set (poset) $P$ has the fixed point property (FPP) if any order preserving function $f:P\longrightarrow P$ has a fixed point.
Theorem. Suppose $P$ and $Q$ are posets with the FPP and at least one of them is finite. Then $(P\times Q)$ has FPP.
Note: $(a,b)\le(c,d)$ if and only if  $a\le c$ and $b\le d$.
Question. Suppose $P$ and $Q$ are two infinite posets with the FPP. Does $(P\times Q)$ have the FPP ?
 A: This an open problem, as far as I know. The finite case was solved by Roddy in 1994. 
There were some more general results proved by other people later:
see this paper by Bernd S. W. Schröder for a survey of history and recent results about FPP in posets. 
A: The question you asked is one of the main long-open problems in fixed point theory of posets.
Commenting on Paul Taylor's interesting answer: the property he describes is called strong fixed point property in order theory. In fact, if one of the posets $P,Q$ has the strong fixed point property and the other has usual fixed point property, then the product $P\times Q$ has the fixed point property. (If both of them have the strong fixed point property, then $P\times Q$ also has the strong fixed point property.) This was proved by Duffus and Sauer in 1980. Fixed point property is not equivalent to strong fixed point property even for finite posets (Pickering, Roddy 1992).
Some approaches that are not mentioned in Schröder's survey mentioned by Gejza Jenča have been presented by Josef Niederle here and here. 
EDIT (made following one of Paul Taylor's comments): Unfortunately all of the papers cited above are behind a paywall. However, one can find free conference versions of Niederle's  work: here and here.
A: Strengthening the hypotheses on $P$ and $Q$ from a fixed point property (every endofunction has some fixed point) to the existence of a fixed point operator $\mathsf{fix}$ with
$$ \text{for any } f:P\to P, \qquad f({\mathsf{fix}}(f)) = {\mathsf{fix}}(f), $$
there is a standard theorem in domain theory or lambda calculus (theoretical computer science) due to Hans Beki&cacute;.
We are given $h:P\times Q\to P\times Q$.  Define
$$ f = \lambda x.{\mathsf{fix}}_P(\lambda y.\pi_1(h(x,y))) : P\to Q $$
$$ g = \lambda y.{\mathsf{fix}}_Q(\lambda x.\pi_0(h(x,y))) : Q\to P $$
which have the properties that
$$ f x = \pi_1(h(x,f x))  \quad\text{and}\quad g y = \pi_0(h(g y,y)). $$
Now let $x_0 = {\mathsf{fix}}_P(\lambda x.g(f x))$ and $y_0=f(x_0)$, so $x_0=g(y_0)$.
Then
$$ \pi_0(h(x_0,y_0)) = \pi_0(h(g y_0,y_0)) = g(y_0) = x_0 $$
$$ \pi_1(h(x_0,y_0)) = \pi_1(h(x_0,f x_0)) = f(x_0) = y_0 $$
so $(x_0,y_0)$ is a fixed point of $h$.
In fact $f$ and $g$ are variables in this, so the argument provides a fixed point operator for $P\times Q$.
Hans Beki&cacute; was a member of the IBM Vienna Laboratory but died in a mountain accident in 1982, leaving a lot of his work unpublished.  It was edited and published by Cliff Jones and is available here. The result above is on pages 38-9 of Bekic2.pdf in this directory.
Google seems to think that this surname has an acute rather than a hachek.
