This problem implies that we cannot have $9$ points. However, we can find $8$ points with the given property as seen below, so the answer is $\boxed{8}$. [asy][asy] dot((0,1),red); dot((0,-1),red); dot((3,1),red); dot((-3,-1),red); dot((1,0),green); dot((-1,0),green); dot((-1,3),green); dot((1,-3),green); draw((0,1)--(0,-1)); draw((0,1)--(3,1)); draw((0,1)--(-3,-1)); draw((0,-1)--(3,1)); draw((0,-1)--(-3,-1)); draw((3,1)--(-3,-1)); draw((1,0)--(-1,0),dotted); draw((1,0)--(1,-3),dotted); draw((1,0)--(-1,3),dotted); draw((-1,0)--(1,-3),dotted); draw((-1,0)--(-1,3),dotted); draw((1,-3)--(-1,3),dotted); [/asy][/asy]