KURSCHAK 2018 Problem 2

Given a prime number $p$ and let $\overline{v_1},\overline{v_2},\dotsc ,\overline{v_n}$ be $n$ distinct vectors of length $p$ with integer coordinates in an $\mathbb{R}^3$ Cartesian coordinate system. Suppose that for any $1\leqslant j<k\leqslant n$ , there exists an integer $0<\ell <p$ such that all three coordinates of $\overline{v_j} -\ell \cdot \overline{v_k}$ is divisible by $p$ . Prove that $n\leqslant 6$ .