Regarding prospects for extending the correspondence to $GL_2(F)$ for other $F$,
one could look at Paškūnas's papers "Coefficient systems and supersingular representations", "Towards a modulo $p$ Langlands correspondence for $GL_2(F)$" (joint with C. Breuil),
and "Admissible unitary completions of locally $\mathbb Q_p$-rational representations of
$GL_2(F)$", available on his website and/or the arXiv.

There is also Breuil's ICM talk from last summer, "The emerging $p$-adic Langlands program", available at his website.
This gives a very nice survey of the whole state of the theory (which has remained relatively stable since then).

Some commentary on Paškūnas's papers: In the $GL_2(\mathbb Q_p)$ case, Breuil found that the numbers of irred. supersingular reps. of $GL_2(\mathbb Q_p)$ mod $p$ matches with the numbers of $2$-dim'l irred. mod $p$ reps. of $G_{\mathbb Q_p}$, and that there is even a natural way to match them (which is e.g. compatible with Serre's conjecture on weights of modular forms giving rise to mod $p$ global Galois reps.).

It was then natural to conjecture that the same was true for $GL_2(F)$.
The first of these papers has the goal of verifying this conjecture. Indeed,
it succeeds in constructing the right number
of supersingular reps. mod $p$ of $GL_2(F)$. However, it was later realized that there was
no way to match these with irred. Galois reps. in any way that is compatible with the Buzzard--Diamond--Jarvis (BDJ) conjecture (the generalization of Serre's conjecture to Hilbert modular forms).

The second paper extends the techniques of the first, and shows in fact that when
$F \neq \mathbb Q_p$ there are *many, many* more supersingulars than there are $2$-dim'l.
irreps of $G_F$. It attempts to find order among this chaos by identifying certain classes of supersingulars which seem to have something to do with the Galois side (in the sense
that they match with the predictions of the BDJ conjecture).

The third paper shows how to lift mod $p$ representations to $p$-adic Banach spaces
representations in interesting ways, and so can be thought of as (i) giving some evidence
that there will be a $p$-adic local Langlands for $GL_2(F)$, but also (ii) showing that
understanding it will be at least as difficult as understanding the mod $p$ situation.