 Solution

At $p$, choose local normal coordinates with $g_{\alpha \beta }(p)=\mathrm{diag}(-1,1)$. Let $e_0,e_1$ be their coordinate basis. The vectors $V=e_0+e_1$ and $W=e_0-e_1$ have zero metric tensor square, and their component matrix has determinant $-2\ne 0$. Thus two independent null vectors form a basis of the tangent space. Normal coordinates are only needed at the chosen point, not on a whole neighbourhood with flat metric.