The consensus interpretation is that FTL is not possible in normal space, since the speed of something is just another way of looking at how "fast" it travels through time - FTL becomes impossible by definition.
However, there is reason to assume a lot of things may be going on that are not strictly speaking taking place in normal flat space.
The article mentions the expansion of the universe as an example, and that's actually a good point to start. By virtue of the universe expanding, the distance between points grows faster than the speed of light as long as those points are sufficiently far apart. This becomes more interesting when you realize that new space in between these points is being created. Sure, the galaxies traveling on this expanding medium are not actually speeding apart powered by classical movement, but they are nevertheless changing position relative to one another in an interesting way.
In fact, what we're interested in when we talk about FTL is not classical movement at all - what we want is to change positions instead. There are places in nature where we know this is happening right now, for example we do know space can "flow" faster than the speed of light on the far side of event horizons.
So if we abandon the idea that we need to accelerate towards a destination, FTL becomes a broader idea of manipulating spacetime in order to be somewhere in less time than if we had actually flown there at the speed of light. There exist proposals for propulsion systems that go in this direction, for example the "warp drive" ideas that cheat by simply shortening the amount of space a vessel has to travel through. Some of these concepts seem to work in theory, for some definition of work.
The trouble with all these ideas is they need impractical amounts of energy. Which is wasteful, since we only need to bend spacetime for a short while and then return it to its original state behind us, something that works analogous to a catalyst would be much better.
At the most fundamental level though, what we are missing is a way of hacking spacetime. Huge masses and energies are how nature bends it, but ideally we would need some other way of manipulating the fabric of existence directly. This is the reason why drive systems in science fiction all have some kind of yet-unknown mystery substance critical to their function. Given how little we actually know about the basic fabric of the universe, ways of doing these manipulations may well exist. They might not. We most likely won't find out for centuries, if ever.
Until then, we must answer the question so that yes, FTL is most likely possible in theory, but it's not even remotely within our grasp right now. It might never work in practice.
Another observation worth considering: cosmic travel times seem utterly problematic right now because we individually live for extremely short periods of time. Spending a few thousand years in transit is much less worrisome if you're, say, an autonomous machine carrying an AI through the galaxy.