Complexity of a decision problem: system of linear equations over finite field with restricted solutions
You can reduce 1-IN-3 SAT to your problem (an instance is a 3CNF, and we want to find a satisfying assignment having exactly one satisfied literal per clause), assuming $p \geq 3$.
A clause $x \lor y \lor z$ is encoded as the constraint $x+y+z=1$.
A clause $\bar x \lor y \lor z$ is encoded as the constraint $1-x+y+z = 1$; and so on.
When $p = 2$, your problem becomes easy.