A $1$-form $\alpha$ over a smooth manifold is non vanishing if for every $p\in M$, $\alpha_p\neq 0$.
But $\alpha_p$ is a linear map $T_p M\to \mathbb R$ hence $\alpha_p(0)=0$. So confusion arises and the precise question is:
What does non vanishing mean for differential forms?
And what does $\alpha\wedge..\wedge\alpha\neq 0$ mean?