Short answer: No. For a smooth proper curve C/k of genus g ≥ 2 over an algebraically closed field of characteristic p > 0, there is currently no nontrivial example for which the full geometric étale fundamental group π1(C) has been determined as an abstract profinite group.

What is known

• Prime-to-p part: For every ℓ ≠ p, the maximal pro-ℓ quotient is the pro-ℓ completion of the topological surface group with one relation (SGA1, Exp. XIII). Equivalently, the maximal prime-to-p quotient π1(C)^(p) is the expected “characteristic 0” surface-group quotient.
• Abelianization: π1(C)ab is canonically isomorphic to the full profinite Tate module of the Jacobian J = Jac(C): π1(C)ab ≅ (∏_{ℓ≠p} T_ℓ(J)) × T_p(J)ét, so the p-primary abelian part has rank equal to the p-rank f of J (0 ≤ f ≤ g). In particular, if f = 0, then π1(C)ab has no p-part.
• Affine curves (for contrast): For open curves, the picture is much better understood; e.g. the maximal pro-p quotient is free pro-p of countable rank (Harbater, Raynaud, Pop), and finite quotients are characterized by the solution of Abhyankar’s conjecture. None of this gives the full group for proper curves.

What remains unknown

• The “wild” pro-p part for proper curves: Beyond the abelianization, essentially nothing explicit is known about the pro-p (or mixed) part of π1(C) for any specific C of genus ≥ 2. Even for very special curves (e.g. supersingular or superspecial curves of small genus), no complete description is known.
• Triviality of the p-part: If J has p-rank 0, then there are no Z/p-quotients (hence no p-group quotients) of π1(C). However, this does not rule out quotients whose order is divisible by p but whose abelianization has no p-part (e.g. non-abelian simple quotients). Whether there exists a curve C with π1(C) equal to its prime-to-p quotient (i.e. with no finite quotient of order divisible by p at all) is not known.

References and context

• SGA1, Exp. XIII: fundamental groups of curves and the pro-ℓ picture (ℓ ≠ p).
• Raynaud and Harbater (and Pop): solution of Abhyankar’s conjecture for affine curves (illustrates the contrast with the proper case).
• Standard descriptions of π1(C)ab via the Jacobian’s Tate modules (e.g. Serre, SGA7 background; any text covering unramified class field theory of curves).

So, apart from the trivial cases g = 0 (π1 = 1) and g = 1 (where π1 is abelian and explicitly the Tate-module-type group above), there is currently no curve of genus ≥ 2 in characteristic p for which the full geometric étale fundamental group has been determined.