= Solution
\b[False.] Vanishing torsion only controls each connected region on which the curvature is nonzero. A smooth curve can pass through a flat point or straight segment and emerge in a different plane.
For an explicit local construction on $(-1,1)$, put $\eta(s)=e^{-1/s^2}$ for $s\ne0$ and $\eta(0)=0$, and define a unit tangent field
$$
T(s)=
\begin{cases}
(\sqrt{1-\eta(s)^2},\eta(s),0),&s<0,\\
(1,0,0),&s=0,\\
(\sqrt{1-\eta(s)^2},0,\eta(s)),&s>0.
\end{cases}
$$
This field is smooth because $\eta$ is flat at zero. Its integral $\alpha(s)=\int_0^sT(u)\,du$ is parametrized by arc length, has nonzero curvature for $s\ne0$, and has zero torsion on both sides. Its negative half lies nontrivially in the $xy$-plane and its positive half lies nontrivially in the $xz$-plane, so the whole curve is not planar.
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