No – at the moment not a single smooth, proper, geometrically connected curve of genus ≥ 2 over an algebraically closed field of characteristic p > 0 is known for which the whole étale fundamental group has been determined.

What is known.

1. Genus 0: π₁(ℙ¹)=1 (the only finite étale covers of ℙ¹ are the trivial ones by the Riemann–Hurwitz formula).

2. Genus 1: For any abelian variety A (in particular for an elliptic curve E) over an algebraically closed field one can classify the finite étale subgroup schemes, hence all finite étale covers, and one obtains

  π₁(A)≅lim← An=∏_{ℓ≠p}T_ℓ(A) × (T_p(A)^{ét}),

 where T_ℓ denotes the ℓ-adic Tate module and T_p(A)^{ét} the étale part of the p-divisible group of A.
For an elliptic curve E this gives
 • ordinary: π₁(E)≅\hat ℤ²×ℤ_p,
 • supersingular: π₁(E)≅\hat ℤ².

These are the only proper curves whose full π₁ is currently understood.

Why nothing is known in genus ≥ 2.

• Prime-to-p part. As you recalled, the prime-to-p quotient is independent of the individual curve:
 π₁(C)^{(p)}≅\widehat{F_{2g}/⟨∏[x_i,y_i]⟩}^{(p)},
so the “mystery’’ is purely in the pro-p part.

• Pro-p part. Even very coarse invariants of the pro-p part are unknown. For instance,
 – we do not know the structure (or even the rank) of the maximal pro-p quotient π₁(C)^{p} for any concrete curve of genus ≥ 2;
 – we cannot write down a presentation;
 – we cannot decide whether it is free pro-p, finitely generated, etc.

• Classification of covers. Determining π₁(C) amounts to classifying all finite étale covers of C. In characteristic p this forces one to understand every unramified Galois extension of the function field k(C) with wildly ramified towers at infinity. Apart from Abhyankar-type existence theorems (showing that π₁ is huge) and various finiteness results (Szpiro, Raynaud, Tamagawa, Pop, …) there is no usable description.

• No special curves behave better. Curves with exceptional automorphism groups (e.g. Hermitian, Suzuki, or superspecial curves), hyperelliptic curves, modular curves, etc., have been studied intensively, yet for none of them is the full profinite fundamental group known.

Consequently, for genus ≥ 2 the complete étale fundamental group is unknown in every concrete example; only partial pieces—prime-to-p quotients, abelianizations, certain solvable or nilpotent quotients—have been described.