A couple of other answers already emphasized that FTL is only supposed to be impossible locally, and that's true whether we're talking about transmitting information or transporting a physical object. This answers the question.
If we drop the "local" caveat, then FTL becomes mundane. In the context of cosmology, this was emphasized in another post:
What does general relativity say about the relative velocities of objects that are far away from one another?
Here I'll mention another example to drive home just how boring FTL becomes if we drop the "local" caveat.
Consider two points $A$ and $B$, both with radial coordinate $r=3M$ in Schwarzschild spacetime (in the usual coordinate system). At this radius, a circular orbit is possible only for something that is moving at the speed of light. Now, suppose that the angular coordinates of these two points are $\phi_A=0$ and $\phi_B=2\pi/1000$, respectively. We can send a circlarly-orbiting light pulse from $A$ to $B$ in either of two directions: the short way (increasing $\phi$), or the long way around the black hole (decreasing $\phi$). If light-pulses $1$ and $2$ both leave $A$ at the same time but pulse $1$ goes the short way and pulse $2$ goes the long way, then pulse $1$ arrives at $B$ before pulse $2$ does. This is an invariant statement, true for all observers. This is (boring) FTL, because pulse $1$ arrived before pulse $2$ even though pulse $2$ was traveling at the speed of light the whole time.
Here's the point: if we disregard the all-important "local" caveat in the usual statement that information/objects can't be transported FTL, then we don't need anything as exotic as a wormhole to achieve FTL.
This answer was written under the assumption that the spacetime does not contain any closed timelike curves. According to Alcubierre's report (https://arxiv.org/abs/gr-qc/0009013), the Alcubierre Drive spacetime does not contain any closed timelike curves. In such a spacetime, causality is safe despite the (mundane) type of FTL described above.
In contrast, the Kerr metric for an eternal rotating black hole does contain closed timelike curves (hidden behind the event horizon), and that does mess with causality — at least it would if we continued to trust general relativity under those conditions, which might not be the right thing to do.
Related: How does "warp drive" not violate Special Relativity causality constraints?