By Noether normalisation any affine variety admits a finite map $f$ to affine space. Hence it is enough to find an example of an affine variety for which the constant sheaf is not perverse. (In this case $f^* \mathbb{Q}_{\mathbb{A}^n}[n]$ will provide a counter-example.)
Now it is easy to find examples where the constant sheaf is not perverse. For example any affine cone $C_X$ over a smooth projective variety $X$ provides a counter-example as long as the cohomology of $C_X \setminus \{ 0 \}$ doesn't look like that of a sphere. So I guess the cone over an elliptic curve would do it.
Edit (...not quite:) see Daniel's comment below.