  Let $P_1,\dots,P_m\in\mathbb{Z}[y]$ be any linearly independent polynomials
with zero constant term. We show that there exists a $\gamma>0$ such that any
subset of $\mathbb{F}_q$ of size at least $q^{1-\gamma}$ contains a nontrivial
polynomial progression $x,x+P_1(y),\dots,x+P_m(y)$, provided the characteristic
of $\mathbb{F}_q$ is large enough.
