Let $F$ be a finite field of size $q$, and let $d<q$. It is well known that for every sequence $x_0,x_1,\ldots, x_{d+1}$ of distinct elements in $F$, there is a sequence $a_1,\ldots,a_d$ of interpolation coefficients such that $f(x_0)=\sum_{i=1,\ldots,d+1}a_i f(x_i)$ for every polynomial function $f$ on $F$ of degree at most d. For a general function $f$ on $F$ that satisfies the above equality, we say that $f$ interpolates well on $x_0,x_1,\ldots, x_{d+1}$.
It is quite easy to verify that a function $f$ that interpolates well on all sequences of distinct elements must be a polynomial function of degree at most $d$. It is also not difficult to see that if $f$ interpolates well on the arithmetic progressions, then it must be a polynomial function of degree at most $d$, in the case where $F$ is a prime field.
My question is: is this still true when the size of the field, $q$, is a prime power?
[Also, is there a more general characterization of sets of interpolation sequences which imply low-degree?]