<B>Edit.</B> I realized after the original post that there are even easier examples. These examples also show that the Quillen-Suslin Theorem fails already for smooth affine quadric hypersurfaces (in some sense, the "simplest" smooth affine varieties after affine spaces, where Quillen-Suslin does hold).
Let $\overline{B}$ be a Grassmannian parameterizing $2$-dimensional quotient vector spaces of a fixed vector space $V$ of dimension $n\geq 4$, $$\overline{B}=\text{Grass}_k(V,2),$$ let the rank $2$, locally free $\mathcal{O}_{\overline{B}}$-module be the tautological quotient, $$V\otimes_k \mathcal{O}_{\overline{B}} \twoheadrightarrow \mathcal{E},$$ and let $D\subset \overline{B}$ be a general hyperplane section for the Plücker embedding. For the open complement $B=\overline{B}\setminus D$, all of the arguments in the original post still apply. Moreover, by van Kampen's theorem, the (tame) fundamental group of $B$ is trivial. So that is definitely a better example than in the original post. For completeness, I include below the original post.
<B>Original post.</B>
Sorry for the delay. I had an overly elaborate example starting with an Enriques surface in characteristic $2$. In fact, sufficiently general K3 surfaces in characteristic $0$ give even simpler examples.
Let $\overline{B}$ be a K3 surface over $\mathbb{C}$. Recall by Mumford's Theorem that the group $\text{CH}_0(\overline{B})$ is uncountably generated, in fact, it cannot even be parameterized by any (finite dimensional) algebraic variety, in an appropriate and precise sense.
MR0249428 <br>
Mumford, D. <br>
Rational equivalence of 0-cycles on surfaces. <br>
J. Math. Kyoto Univ. 9 (1968), 195–204. <br>
https://projecteuclid.org/journals/journal-of-mathematics-of-kyoto-university/volume-9/issue-2/Rational-equivalence-of-0-cycles-on-surfaces/10.1215/kjm/1250523940.full
Moreover, in every ample divisor class on $\overline{B}$, there exists a divisor $D$ whose irreducible components have normalization isomorphic to $\mathbb{P}^1$, cf. Beauville's exposition of the Yau-Zaslow theorem.
MR1682284 <br>
Beauville, Arnaud <br>
Counting rational curves on K3 surfaces. <br>
Duke Math. J. 97 (1999), no. 1, 99–108. <br>
https://arxiv.org/pdf/alg-geom/9701019.pdf
In fact, by a theorem of Beauville-Voisin, the subgroup of $\text{CH}_0(\overline{B})$ generated by all pushforwards of all $0$-cycles on (possibly singular) rational curves in $\overline{B}$ is a cyclic subgroup commensurate with the cyclic subgroup generated by the cycle class $c_2(T_{\overline{B}})$ and containing all cup products of first Chern classes of divisors.
MR2047674 <br>
Beauville, Arnaud; Voisin, Claire <br>
On the Chow ring of a K3 surface. <br>
J. Algebraic Geom. 13 (2004), no. 3, 417–426. <br>
https://arxiv.org/abs/math/0111146
Now assume that $\overline{B}$ is a very general K3 surface, so that the Picard group is generated by an ample $\mathcal{O}_{\overline{\mathcal{B}}}$-module $\mathcal{L}$. For definiteness, assume that the degree of $c_1(\mathcal{L})\cap c_1(\mathcal{L})$ equals $2$.
Let $D$ be an irreducible, nodal curve in the linear system of $\mathcal{L}$ such that the normalization of $D$ is a rational curve, i.e., an irreducible, nodal curve of arithmetic genus $2$ and geometric genus $0$. For the affine surface $B=\overline{B}\setminus D$, there exists a closed point $p\in B$ such that the cycle class of $\{p\}$ in $\text{CH}_0(\overline{B})$ is not contained in the cyclic subgroup coming from pushforwards of $0$-cycles from $D$. In particular, the cycle class of $\{p\}$ in $\text{CH}_0(B)$ does not equal a cup product of two divisor classes.
Apply Serre's construction to the ideal sheaf $\mathcal{I}_p\subset \mathcal{O}_{\overline{B}}$ with determinant invertible sheaf $\mathcal{L}=\mathcal{O}_{\overline{B}}(D)$, i.e., consider the local-to-global spectral sequence computing $\text{Ext}^*_{\mathcal{O}_{\overline{B}}}(\mathcal{I}_p,\mathcal{L})$. The long exact sequence of low degree terms is $$ 0 \to H^1(\overline{B},\mathcal{L}) \to \text{Ext}^1_{\mathcal{O}_{\overline{B}}}(\mathcal{I}_p,\mathcal{L}) \to L|_p \to H^2(\overline{B},\mathcal{L}) .$$
Here $L|_p$ is the $1$-dimensional vector space that is, essentially, the fiber at $p$ of $\mathcal{L}$. Since the dualizing sheaf on $\overline{B}$ is isomorphic to the structure sheaf, Kodaira vanishing gives that $H^q(\overline{B},\mathcal{L})$ is the zero vector space for $q=1,2$. Thus, there is a unique isomorphism class of a nontrivial extension $\mathcal{E}$ of $\mathcal{I}_p$ by $\mathcal{L}$.
The sheaf $\mathcal{E}$ is a locally free $\mathcal{O}_{\overline{B}}$-module of rank $2$ whose first Chern class equals $c_1(\mathcal{L})$ and whose second Chern class equals the cycle class of $\{p\}$. In particular, the restriction $\mathcal{E}|_B$ is a locally free $\mathcal{O}_B$-module of rank $2$ whose first Chern class is zero and whose second Chern class is the nonzero class of $\{p\}$.
Consider the group scheme $G$ over $\overline{B}$ parameterizing $\mathcal{O}_B$-automorphisms of $\mathcal{E}$ such that the induced automorphism of $\text{det}(\mathcal{E}) \cong \mathcal{L}$ is the identity automorphism. This is an inner form of $\textbf{SL}_2$ over $\overline{B}$. Since $\mathcal{E}$ is locally free, this inner form is Zariski locally isomorphic to $\textbf{SL}_2$, and thus has split maximal tori Zariski locally. However, the restriction over $B$ does not have a maximal torus. If it did, the corresponding representation $\mathcal{E}|_B$ would split as a direct sum of locally free sheaves of rank $1$. Then, by the Whitney sum formula, the second Chern class of $\mathcal{E}|_B$ would be in the image of the cyclic subgroup of Beauville-Voisin, i.e., the second Chern class would be a torsion class in $\text{CH}_0(B)$, contrary to the choice of $p$.
Now consider the projective space bundle over $\overline{B}$ that parameterizes invertible quotients of the pullback of $\mathcal{E}$. This is a projective scheme. By Bertini theorems, an intersection of this projective scheme with a sufficiently general hyperplane is a smooth closed subscheme $\overline{A}$ whose projection to $\overline{B}$ is finite surjective, and thus finite flat. In particular, the restriction $A$ over $B$ is a finite, flat cover of $B$. The pullback of $\mathcal{E}|_B$ to $A$ has an invertible quotient. Since $A$ is affine, thus surjection automatically splits, so that the pullback of $\mathcal{E}|_B$ to $A$ is isomorphic to a direct sum of two invertible sheaves. Thus, the pullback of $G$ to $A$ does have a maximal torus.