Let $X$ be a complete intersection in $\mathbb{P}^n$ of multidegree $(d_1,\ldots,d_r)$. If we're working over a finite field $\mathbb{F}_q$, the Ax-Chevalley-Warning theorem says that if $X$ is in the Fano range, i.e., $$ \sum d_i \leq n,$$ then $|X(\mathbb{F}_q)| \equiv 1 \pmod q.$

In the case that $X$ is not in the Fano range, one can cook up examples of such $X$ with no $\mathbb{F}_q$-points at all using the norm form, but these are not smooth.

In SGA 7 II Exposè XXI, Katz shows that for a general complete intersection in the Fano range, this congruence is not satisfied, possibly after extending the ground field.

Given a multidegree outside the Fano range, are there explicit examples of smooth complete intersections which don't satisfy this congruence?

Edit: Sorry, I wasn't sufficiently explicit about what I was asking. Restricting to hypersurfaces, what I would like is for each $d$, $p$, and $n$ with $d \geq n+1$, an explicit example of a smooth hypersurface of degree $d$ in $\mathbb{P}^n$ which doesn't satisfy the Chevalley-Warning congruence.

Thanks for the examples, though; I knew some explicit examples before but no infinite families.