Suppose that $X\subset\mathbb P^n$ is a $d$-dimentional smooth projective variety (not a linear subspace) over an algebraically closed field. If $\gamma\colon X\to\mathrm{Gr}(d,\mathbb P^n)$ is Gauss mapping that attaches to each point $x\in X$ the embedded Zariski tangent space to $X$ at $x$, then it is known that $\gamma$ is finite.

If characteristic is zero, it is known that $\gamma$ is not just finite but birational onto its image. My question is whether $\gamma$ is generically one to one in finite characteristic.

Edit: removed the question about birationality in finite characteristic, thanks to the example given by Felipe Voloch.

Thanks in advance,

Serge