Somebody answered this question instead of the question here, so I am asking this with the hope that they will cut and paste their solution.
[Math] an example of a smooth variety over a finite field F_p which does not lift to Z_p
ag.algebraic-geometrydeformation-theoryfinite-fields
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EDIT 7/15/14 I was just looking back at this old answer, and I don't think I ever answered the stated question. I can't delete an accepted answer, but I'll point at that, as far as I can tell, the Vakil reference I give also only address the question of deforming $X$ over $\mathbb{Z}_p$, not of embedding it in some larger flat family over $\mathbb{Z}_p$.
EDIT Oops! David Brown points out below that I misread the question. I was answering the question of finding a smooth scheme which does not deform in a smooth family over Z_p.
Well, to make up for that, I'll point to some references which definitely contain answers. Look at section 2.3 of Ravi Vakil's paper Murphy's Law in algebraic geometry: Badly-behaved deformation spaces for some history, and several good references. Moreover, Ravi describes how to build an explicit cover of P^2 in characteristic p which does not deform to characteristic 0. Basically, the idea is to take a collection of lines in P^2 which doesn't deform to characteristic 0 and take a branched cover over those lines. For example, you could take that p^2+p+1 lines that have coefficients in F_p.
Using some of BCnrd's ideas together with a different construction, I'll give a positive answer to Kevin Buzzard's stronger question; i.e., there is a counterexample for any non-etale smooth morphism.
Call a morphism $X \to Y$ wicked smooth if it is locally of finite presentation and for every (square-zero) nilpotent thickening $T_0 \subseteq T$ of $Y$-schemes, every $Y$-morphism $T_0 \to X$ lifts to a $Y$-morphism $T \to X$.
Theorem: A morphism is wicked smooth if and only if it is etale.
Proof: Anton already explained why etale implies wicked smooth.
Now suppose that $X \to Y$ is wicked smooth. In particular, $X \to Y$ is smooth, so it remains to show that the geometric fibers are $0$-dimensional. Wicked smooth morphisms are preserved by base change, so by base extending by each $y \colon \operatorname{Spec} k \to Y$ with $k$ an algebraically closed field, we reduce to the case $Y=\operatorname{Spec} k$. Moreover, we may replace $X$ by an open subscheme to assume that $X$ is etale over $\mathbb{A}^n_k$ for some $n \ge 0$.
Fix a projective variety $P$ and a surjection $\mathcal{F} \to \mathcal{G}$ of coherent sheaves on $P$ such that some $g \in \Gamma(P,\mathcal{G})$ is not in the image of $\Gamma(P,\mathcal{F})$. (For instance, take $P = \mathbb{P}^1$, let $\mathcal{F} = \mathcal{O}_P$, and let $\mathcal{G}$ be the quotient corresponding to a subscheme consisting of two $k$-points.) Make $\mathcal{O}_P \oplus \mathcal{F}$ an $\mathcal{O}_P$-algebra by declaring that $\mathcal{F} \cdot \mathcal{F} = 0$, and let $T = \operatorname{\bf Spec}(\mathcal{O}_P \oplus \mathcal{F})$. Similarly, define $T_0 = \operatorname{\bf Spec}(\mathcal{O}_P \oplus \mathcal{G})$, which is a closed subscheme of $T$ defined by a nilpotent ideal sheaf. We then may view $g = 0+g \in \Gamma(P,\mathcal{O}_P \oplus \mathcal{G}) = \Gamma(T_0,\mathcal{O}_{T_0})$.
Choose $x \in X(k)$; without loss of generality its image in $\mathbb{A}^n(k)$ is the origin. Using the infinitesimal lifting property for the etale morphism $X \to \mathbb{A}^n$ and the nilpotent thickening $P \subseteq T_0$, we lift the point $(g,g,\ldots,g) \in \mathcal{A}^n(T_0)$ mapping to $(0,0,\ldots,0) \in \mathbb{A}^n(P)$ to some $x_0 \in X(T_0)$ mapping to $x \in X(k) \subseteq X(P)$. By wicked smoothness, $x_0$ lifts to some $x_T \in X(T)$. The image of $x_T$ in $\mathbb{A}^n(T)$ lifts $(g,g,\ldots,g)$, so each coordinate of $x_T$ is a global section of $\mathcal{F}$ mapping to $g$, which is a contradiction unless $n=0$. Thus $X \to Y$ is etale.
Best Answer
Examples are also in my paper "Murphy's Law in Algebraic Geometry", which you can get from my preprints page
Here is a short (not quite complete) description of a construction, with two explanations of why it works. I hope I am remembering this correctly!
In characteristic $>2$, consider the blow up of $\mathbf{P}^2$ at the $\mathbf{F}_p$-valued points of the plane. Take a Galois cover of this surface, with Galois group $(\mathbf{Z}/2)^3$, branched only over the proper transform of the lines, and the transform of another high degree curve with no $\mathbf{F}_p$-points. Then you can check that this surface violates the numerical constraints of the Bogomolov-Miyaoka-Yau inequality, which holds in characteristic zero; hence it doesn't lift. (This is in a paper by Rob Easton.) Alternatively, show that deformations of this surface must always preserve that Galois cover structure, which in turn must preserve the data of the branch locus back in $\mathbf{P}^2$, meaning that any deformation must preserve the data of those $p^2+p+1$ lines meeting $p+1$ to a point, which forces you to live over $\mathbf{Z}/p$.
The two papers mentioned above give more exotic behavior too (of different sorts in the two papers), e.g. you an find a surface that lifts to $\mathbf{Z}/p^{10}$ but still not to $\mathbf{Z}_p$.