Consider a set of Cartesian coordinates $x,y,z$, and a force field which points only in the direction of $z$: $\vec{F} = (0,0,F_z)$. Now consider a particle with a general velocity $\vec{v}$, we can always rotate the coordinate system around the $z$ axis so that the $x$ component of velocity is eliminated and we have $\vec{v} = (0,v_y,v_z)$. Now the equations of motion read

$$\ddot{z} = \frac{F_z}{m},\; \ddot{x} = \ddot{y} = 0$$

That is, $v_y = const.$ and $v_x = const$. Obviously, this particle will then stay in the $y-z$ plane and never move in the $x$ direction. This means that the motion is planar and this is a counterexample to your question.