Now that our paper Geometrization of the local Langlands correspondence with Fargues is finally out (ooufff!!), it may be worth giving an update to Ben-Zvi's answer above. In brief: we give a formulation of Local Langlands over a $p$-adic field $F$ so that it is finally
- an actual conjecture, in the sense that it asks for properties of a given construction, not for a construction;
- of a form as in geometric Langlands, in particular about an equivalence of categories, not merely a bijection of irreducibles.
First, I should say that in the notation of the OP, we construct a canonical map $\Pi(G)\to \Phi(G)$, and prove some properties about it. However, we are not able to say anything yet about its fibres (not even finiteness).
Moreover, we give a formulation of local Langlands as an equivalence of categories, and (essentially) construct a functor in one direction that one expects to realize the equivalence. In particular, this nails down what the local Langlands correspondence should be, it "merely" remains to establish all the desired properties of it.
Let me briefly state the main result here. Let $\mathrm{Bun}_G$ be the stack of $G$-bundles on the Fargues--Fontaine curve. We define an ($\infty$-)category $\mathcal D(\mathrm{Bun}_G,\overline{\mathbb Q}_\ell)$ of $\ell$-adic sheaves on $\mathrm{Bun}_G$. The stack $\mathrm{Bun}_G$ is stratified into countably many strata enumerated by $b\in B(G)$, and on each stratum, the category $\mathcal D(\mathrm{Bun}_G^b,\overline{\mathbb Q}_\ell)$ is the derived ($\infty$-)category of smooth representations of the group $G_b(F)$. In particular, for $b=1$, one gets smooth representations of $G(F)$.
Moreover, there is an Artin stack $Z^1(W_F,\hat{G})/\hat{G}$ of $L$-parameters over $\overline{\mathbb Q}_\ell$.
Our main result is the construction of the "spectral action":
There is a canonical action of the $\infty$-category of perfect complexes on $Z^1(W_F,\hat{G})/\hat{G}$ on $\mathcal D(\mathrm{Bun}_G,\overline{\mathbb Q}_\ell)$.
The main conjecture is basically that this makes $\mathcal D(\mathrm{Bun}_G,\overline{\mathbb Q}_\ell)^\omega$ a "free module of rank $1$ over $\mathrm{Perf}(Z^1(W_F,\hat{G})/\hat{G})$", at least if $G$ is quasisplit (or more generally, has connected center).
More precisely, assume that $G$ is quasisplit and fix a Borel $B\subset G$ and a generic character $\psi$ of $U(F)$, where $U\subset B$ is the unipotent radical, giving the Whittaker representation $c\text-\mathrm{Ind}_{U(F)}^{G(F)}\psi$, thus a sheaf on $[\ast/G(F)]$, which is the open substack of $\mathrm{Bun}_G$ of geometrically fibrewise trivial $G$-bundles; extending by $0$ thus gives a sheaf $\mathcal W_\psi\in \mathcal D(\mathrm{Bun}_G,\overline{\mathbb Q}_\ell)$, called the Whittaker sheaf.
Conjecture. The functor
$$ \mathrm{Perf}(Z^1(W_F,\hat{G})/\hat{G})\to \mathcal D(\mathrm{Bun}_G,\overline{\mathbb Q}_\ell)$$
given by acting on $\mathcal W_\psi$ is fully faithful, and extends to an equivalence
$$\mathcal D^{b,\mathrm{qc}}_{\mathrm{coh}}(Z^1(W_F,\hat{G})/\hat{G})\cong \mathcal D(\mathrm{Bun}_G,\overline{\mathbb Q}_\ell)^{\omega}.$$
Here the superscript $\mathrm{qc}$ means quasicompact support, and $\omega$ means compact objects. As $Z^1(W_F,\hat{G})$ is not smooth (merely a local complete intersection), there is a difference between perfect complexes and $\mathcal D^b_{\mathrm{coh}}$, and there is still a minor ambiguity about how to extend from perfect complexes to all complexes of coherent sheaves. Generically over the stack of $L$-parameters, there is however no difference.
It takes a little bit of unraveling to see how this implies more classical forms of the correspondence, like the expected internal parametrization of $L$-packets; in the case of elliptic $L$-parameters, everything is very clean, see Section X.2 of our paper.
(There are related conjectures and results by Ben-Zvi--Chen--Helm--Nadler, Hellmann and Zhu; see also the work of Genestier--Lafforgue in the function field case. And this work is heavily inspired by previous work in geometric Langlands, notably the conjectures of Arinkin--Gaitsgory, and the work of Nadler--Yun and Gaitsgory--Kazhdan--Rozenblyum--Varshavsky on spectral actions.)
PS: It may be worth pointing out that this conjecture is, at least a priori, of a quite different nature than Vogan's conjecture, mentioned in the other answers, which is based on perverse sheaves on the stack of $L$-parameters; here, we use coherent sheaves.
Best Answer
I am not sure it makes sense to ask "what is the p-adic local Langlands conjecture for $\mathrm{GL}_1$". Nobody has succeeded in even formulating a reasonable candidate for a p-adic LLC for $\mathrm{GL}_n$, so you can't just plug $n = 1$ in and see what it says. Morally, the conjecture should be something like this:
The problem is: what properties? Saying "there is a bijection" is just claiming that the sets have the same cardinality, which is trivial, and clearly not what is intended here (Serre makes exactly this point in his lecture How To Write Mathematics Badly, which is on YouTube). So one needs to spell out what the properties are, and this is hard to do.
In this $\mathrm{GL}_1$ setting you can cheat, because there is already a candidate for this bijection (any continuous unitary character $K^\times \to E^\times$ extends uniquely to the profinite completion, and $\widehat{K^\times}$ is the abelianization of $\mathrm{Gal}(\bar{K} / K)$ by class field theory). But one doesn't know how to uniquely pin down what the question is for $\mathrm{GL}_n$, which is one reason why it's hard to answer!
Even for $\mathrm{GL}_2 / \mathbf{Q}_p$, Colmez has constructed a bijection between p-adic Banach reps of $\mathrm{GL}_2(\mathbf{Q}_p)$ and 2-dimensional Galois reps, which has a long list of very nice properties; but as far as I'm aware there's not a theorem which says "Colmez's correspondence is the unique correspondence with properties X, Y and Z".
EDIT. Let me add some more about the global side. Here it's even less clear what the question is. For $\mathrm{GL}_n$ over a number field $K$, the statement should probably be that there should be a bijection between: $n$-dimensional $p$-adic representation of $\mathrm{Gal}(\overline{K} / K)$ which are continuous and unramified almost everywhere; and "p-adic automorphic representations of $\mathrm{GL}_n$". But it's not at all clear what the definition of the latter class of objects should be for $n \ge 2$.
The eigenvariety for $\mathrm{GL}_1 / K$, $K$ a number field, is rather well understood. I'll just summarize Buzzard's construction briefly. Choose some open compact subgroup $U$ of $\left(\mathbf{A}_K^{(p\infty)}\right)^\times$ (the finite adeles away from $p$). Let $K_\infty^\circ$ be the totally positive elements of $(K \otimes \mathbf{R})^\times$. Then the quotient $$ G(U) = \mathbf{A}_K^\times / \overline{ U \cdot K_\infty^\circ \cdot K^\times } $$ is a pretty friendly abelian group; it has a subgroup of finite index isomorphic to $\mathbf{Z}_p^n$ for some integer $1 \le n \le [K : \mathbf{Q}]$. Moreover, Leopoldt's conjecture is very easily seen to be equivalent to determining $n$ (it says that $n = 1 + r_2$, where $r_2$ is the number of complex places of $K$). Since $G(U)$ is such a friendly group, it's easy to see that $p$-adic characters of $G(U)$ naturally biject with points of a rigid space $\mathcal{E}(U)$ and that's your eigenvariety.
Now, the points of $\mathcal{E}(U)$ can be viewed as continuous $p$-adic characters of $\mathbf{A}_K^\times / K^\times$ with prime-to-$p$ ramification bounded by $U$, or (via class field theory) as 1-dimensional Galois representations, again with prime-to-$p$ ramification bounded by $U$. That gives a bijection between these classes of objects, to which it seems natural to give the name "p-adic Langlands for $GL_1$". So there's the relation between the objects in Chandan's question:
As soon as you step away from $\mathrm{GL}_1$, though, eigenvarieties get less central to the $p$-adic Langlands picture (although they're still clearly very important). The issue now is that eigenvarieties are expected to parametrise Galois representations which are fairly "degenerate" locally at $p$: technically they should be trianguline, a notion introduced by Colmez. For $\mathrm{GL}_1$ everything is (vacuously) trianguline, but this is further and further away from being true as $n$ grows; e.g. a Galois representation with finite image is only trianguline if it's a direct sum of characters. Trianguline representations should correspond under p-adic Langlands to representations of $\mathrm{GL}_n$ induced from the Borel subgroup -- so in a sense these are the easiest ones!