The thing about the speed of light $c$ is that it's not just a number associated with a certain type of particle. While we could talk about the mass of the proton, and there would be no problem assuming non-protons had greater or lesser masses, the value $c$ is an entirely different beast.
$c$ is an intrinsic property of spacetime itself, not of the particles in spacetime. You wouldn't expect there to exist anything (particle, signal, information) that you could insert into spacetime with the property "changes the nature of spacetime for itself."
The above is meant to undermine the "if it's just a property of everything we've found so far, we haven't ruled out finding things without this property" line of thought.
For a more concrete demonstration of what goes wrong when you alter the structure of spacetime as we know it, take a look at the "tachyonic antitelephone". This demonstrates how any abstract communication faster than light leads to causality violations as observed by even not-faster-than-light observers.
There are many variations on the thought experiment; here's one of them (with the math worked out in the linked article): $A$ is moving away from $B$ with speed $v < c$ in $B$'s reference frame. $A$ sends a faster-than-light message to $B$, who responds in kind with a reply. The problem is for sufficiently fast (but not faster than $c$!) speeds $v$, $A$ will receive the reply before sending the message. Faster-than-light anything begets time travel.
Since it gets repeated far, far too often, I'll also counter quantum entanglement arguments here. Quantum entanglement does nothing in the hypothetical "why can't this communicate faster than light?" scenarios other than guarantee that the two particles will "collapse" to the same (or opposite, or orthogonal, or whatever) states when either is observed. The only quantum mechanical aspect of the whole thing is the fact that the eventual collapsed state can't even exist before the measurement (it's not a hidden variable).
But the correlation -- the thing you want to rely on to communicate faster than light -- could be achieved entirely classically. Take a red marble and a blue marble, put them in a bag, and draw one out randomly without looking. Lock your marble in a box, and hand the bag to someone else, who also doesn't peak. Send the person far away. Then, look at your marble. If it's blue, you instantly know the other person has red. But you transmitted nothing. All quantum mechanics does is make it so who has which marble isn't pre-determined.
This shouldn't come as a surprise, because the only notions of space and time in quantum mechanics are the ones from spacetime itself. Quantum mechanics doesn't come equipped with some independent notion of distances. So whatever structure spacetime has applies to quantum things just as much as marbles and people and signals.
Everything said here stays true even in general relativity, by the way. Enabling things to move faster than $c$ still leads to causality violations. Even wrapping things in a "warp bubble" and moving that faster than $c$ leads to causality violations.