Is a surface curve made of planar points necessarily a line?

I am having trouble proving the following statement, which at first seemed intuitively true to me.

Let $S$ be a surface in $mathbb{R}^{3}$. Suppose that there is a curve $gamma$ in $S$ whose all points are planar, i.e., the second fundamental form $alpha$ (or, equivalently, the shape operator) of $S$ vanishes at all points of $gamma$. Does this imply that $gamma$ is part of a straight line?

This question is related to the existence of non-straight asymptotic curves in $S$. It is well-known that a curve $gamma$ such that $alpha(dot{gamma},dot{gamma})=0$ need not be part of a straight line.

EDIT: As pointed out by Arctic Char, the claim is not true in general. What happens if we assume that, for every open neighborhood $U$ (in $S$) of the curve $gamma$, there is no plane $P$ such that $U subset P$?