Non-algebraic K3 surfaces in characteristic $p$ I have a very naive question.
Recall that over the field of complex numbers, there exist non-algebraic K3 surfaces. Namely, smooth non-projective simply connected compact complex surfaces with trivial canonical bundle. Such surfaces often come in useful when studying the moduli and deformations of K3 surfaces over $\mathbb{C}$.

Is there a notion of "non-algebraic" K3 surfaces over fields of positive characteristic?

As I say above, I am well aware that this question is very naive. Possible guesses I had in mind were formal schemes / stacks / rigid analytic spaces / ...?
 A: Let me briefly expand on Jason's comment.
Actually, "formal scheme" is the right word here. 
For any K3 surface $X$ over an algebraically closed field $k$ of any characteristic one has $$h^0(X, T_X)=0, \quad  h^1(X, T_X)=20, \quad \, h^2(X, T_X)=0,$$
so the functor of Artin rings $$F \colon  (\textbf{Art}) \to (\textbf{Sets})$$  describing the first-order deformations of $X$ is pro-representable and unobstructed, hence it is represented by a complete, regular local ring $A$ of dimension $20$. More precisely, $$A \cong  k[[x_1, \ldots, x_{20}]],$$
where $x_1, \ldots, x_{20}$ are indeterminates. 
Thus there is a formal universal deformation $\widehat{\mathcal{X}} \to \textrm{Specf} (A)$ of $X$; however, such a deformation is not effective, in the sense that there exists no algebraic deformation $\mathcal{X} \to \textrm{Spec}(A)$, where $ \mathcal X$ is a scheme, such that $\widehat{\mathcal{X}}$ is the formal completion of $\mathcal{X}$ along the closed fibre $X$. In fact, a straightforward cup product computation shows that any algebraic family of K3 surfaces with injective Kodaira-Spencer map at any point has dimension $\leq 19$.
Deligne showed that any K3 surface over $k$ can be lifted to $\mathbb{C}$. So there is an analogy with the complex analytic theory, where the deformation space, as a complex manifold, has dimension $20$, but the algebraic K3 surfaces form a (countable union of) $19$-dimensional subfamilies. 
More details on this topic can be found in the books Deformations of algebraic schemes (Sernesi) and Deformation theory (Hartshorne).
