Are there any examples known of an affine variety $A$ over an algebraically closed field such some Chow group (say, of codimension at least $2$) of $A$ with coefficients in $\mathbb{Z}/n\mathbb{Z}$ ($n$ is prime to the residue field characteristic) is nonzero?

I think there will be many such varieties. For example, let $Q$ be a smooth $4$dimensional projective quadric, $Q'$ a hyperplane section of $Q$ and $A = Q \backslash Q'$. For any $i$, we have the localisation sequence $$ CH^{i1}(Q') \to CH^i(Q) \to CH^i(A) \to 0 \ . $$ Since $CH^1(Q') = \mathbb{Z}/n\mathbb{Z}$ and $CH^2(Q) = (\mathbb{Z}/n\mathbb{Z})^{\oplus 2}$ (with $\mathbb{Z}/n\mathbb{Z}$ coefficients) it follows that $CH^2(A)$ is nonzero. 

