Technically we already know how to violate our universes's speed limit by bending space rather than moving through it normally. Trouble is it would require us being able to harness the energy of maybe a dozen suns, and it doesn't really count since you aren't just moving in a straight line like any other means of conveyance.
While we have know the broad strokes of how it can be done (even more Trek tech!) it's not something that's going to be feasible at any point in the forseeable future, and by the time it is I'm sure we'll have better adaptations available anyway. (Such as suspended animation, generational ships, etc.)
But if you're asking in terms of straight up going faster than light, not 'merely' bending space around your ship, the answer is absolutely no unless and until our understanding of the universe is dramatically altered yet again.