I am just posting my comment above as an answer.  Donu's comment is exactly correct: I failed to point out that the argument proves that $H^1_{et}(\mathbb{A}^1_k,\mathbb{Z}/p\mathbb{Z})$ is infinite.

Consider Artin-Schreier extensions of $\mathbb{A}^1_k$, i.e., $\text{Spec}\ k[t][x]/\langle x^p-x-f(t)\rangle$ for $f(t)\in k[t]$.  By the Artin-Schreier sequence, this $\mathbb{Z}/p\mathbb{Z}$-torsor over $\mathbb{A}^1_k$ is classified by the image $[f]$ in the cokerenel of $A:k[t]\to k[t], \ A(g) = g^p-g$.  A $k$-basis for this quotient consists of the image $[t^d]$ of the infinitely many monomials $t^d$ with $d$ relatively prime to $p$.