If the base field $k$ is algebraically closed then a section exists because of a group-theoretic property of the tame fundamental group (considering this question over an algebraically closed field $k$ seems to be the closest analogy to vector bundles with connection over $\mathbb{C}$)
Esnault, Shusterman, and Srinivas proved in Theorem 1.2 here https://arxiv.org/abs/2102.13424 that for an affine curve $U$ over an algebraically closed field $k$ the tame fundamental group $\pi_1^{tame}(U)$ is a projective object of the category of profinite groups. By definition, this means that for any continuous surjection $f:A\twoheadrightarrow B$ of profinite groups, any homomorphism $\pi_1^{tame}(U)\to B$ can be lifted along $f$ to a homomorphism $\pi_1^{tame}\to A$. In particular, the surjection $\pi_1(U)\to \pi_1^{tame}(U)$ has a section.
Note that projectivity of the prime-to-$p$ completion of $\pi_1^{tame}(U)$ (which coincides with the prime-to-$p$ completion of the whole $\pi_1(U)$) is rather classical because, by Grothendieck's specialization and comparison to topological fundamental group, $\pi_1(U)^{prime-to-p}$ is the free prime to $p$ profinite group on finitely many generators. But the authors of this paper prove projectivity of the tame fundamental group as it is, by using a theorem of Shafarevich that the pro-$p$ completion of $\pi_1$ of a smooth projective curve over an algebraically closed field of characteristic $p$ is a free pro-p group (note that when $l\neq p$ this last statement is false for the pro-$l$ completion!).
However, there does not seem to be a canonical choice of the section $\pi_1^{tame}(U)\to \pi_1(U)$. Worse, over a non-closed field $k$ the section need not exist. Before giving a specific example, let me remark that from the point of view of Grothendieck's anabelian conjectures we shouldn't expect such a section to exist for general fields $k$. If $U$ is an affine curve over a finite field $k$ with the smooth compactification $\overline{U}$, then the pro-p-completion of $\pi_1^{tame}(U)$ coincides with the pro-p-completion of $\pi_1(\overline{U})$ because any tamely ramified Galois cover of $p$-power order of $\overline{U}$ has to be unramified. So a section $\pi_1^{tame}(U)\to \pi_1(U)$ would induce a map of pro-p-completions $\pi_1(\overline{U})^{pro-p}\to \pi_1(U)^{pro-p}$. But if we believe that the pro-p fundamental group of a hyperbolic curve over a finite field of characteristic $p$ functorially determines the curve, then this map would have to arise from a non-constant map $\overline{U}\to U$, which is absurd.
To be fair, I'm not sure if it is reasonable to expect that the pro-p completion (rather than the whole group) is enough to determine the hyperbolic curve over a finite field (but it is e.g. over a p-adic field, by a theorem of Mochizuki). In any case, we don't need the full power of this statement to disprove the existence of a section, and here is a self-contained argument:
Let $E$ be an ordinary elliptic curve over a large enough finite field $k$ so that $E(k)[p]\simeq\mathbb{Z}/p$, and take $U=E\setminus \{0\}$. If $\pi_1(U)\to \pi_1^{tame}(U)$ had a section $s:\pi_1^{tame}(U)\to \pi_1(U)$ then (choosing a $k$-rational base point of $U$) it would induce a section on geometric fundamental groups $s_{\overline{k}}:\pi_1^{tame}(U_{\overline{k}})\to \pi_1(U_{\overline{k}})$ equivariant for the absolute Galois group $G_k=Gal(\overline{k}/k)$. This would imply that the injection
$$\mathrm{Hom}(\pi_1^{tame}(U_{\overline{k}}),\mathbb{Z}/p)\to \mathrm{Hom}(\pi_1(U_{\overline{k}}),\mathbb{Z}/p)$$
induced by the surjection $\pi_1(U_{\overline{k}})\to \pi_1^{tame}(U_{\overline{k}})$ admits a $G_k$-equivariant retraction. Since any tame Galois cover of $U_{\overline{k}}$ of $p$-power order has to extend to an étale cover of the compactification $E_{\overline{k}}$, the above injection can be rewritten as the restriction map on étale cohomology:
$$H^1_{\text{ét}}(E_{\overline{k}},\mathbb{Z}/p)\to H^1_{\text{ét}}(U_{\overline{k}},\mathbb{Z}/p)$$
By out assumption that $E$ is ordinary with all of its $p$-torsion defined already over $k$, the cohomology group $H^1_{\text{ét}}(E_{\overline{k}},\mathbb{Z}/p)$ is $\mathbb{Z}/p$ with trivial Galois action. The cohomology of $U_{\overline{k}}$ is described as
$$H^1_{\text{ét}}(U_{\overline{k}},\mathbb{Z}/p)=\mathrm{coker}(\mathrm{Fr}_p-1:\mathcal{O}(U)\otimes_k \overline{k})$$
by the Artin-Schreier sequence, where $G_k$ acts on it through the action on $\overline{k}$. If the injection $\mathbb{Z}/p=H^1_{\text{ét}}(E_{\overline{k}},\mathbb{Z}/p)\to H^1_{\text{ét}}(U_{\overline{k}},\mathbb{Z}/p)$ had a Galois-equivariant retraction, then, in particular, coinvariants of the action of $G_k$ on $H^1_{\text{ét}}(U_{\overline{k}},\mathbb{Z}/p)$ would be non-zero. By this is not the case, because coinvariants of $G_k$ acting on $\overline{k}$ are zero by the standard vanishing $H^1(G_k,\overline{k})=0$.
Lastly, such a section exists locally around every puncture, if $k$ is a finite field. The surjection of Galois groups $G_{k((t))}\to G^{tame}_{k((t))}$ admits a section for a rather soft group-theoretic reason: the group $G^{tame}_{k((t))}$ is an extension of $\widehat{\mathbb{Z}}$ by a prime-to-p profinite group, hence any extension of it by a pro-p group admits a section. I believe this argument is originally due to Iwasawa, see Theorem 7.5.3 in Neukirch's book 'Cohomology of Number Fields' for a more detailed exposition.
