In Newtonian mechanics, it would be possible, but unfortunately, it does not provide a valid description for what happens to massive objects at speeds approaching c.

To define a feasible, closed system in general relativity for massive objects travelling at c wouldn't work as the kinetic energy required would be infinite. Conservation of mass would still apply for objects with speeds exceeding c from some inertial frames (e.g. objects travelling through wormholes, although technically, ripples in spacetime can't normally occur in general relativity either), but you wouldn't be able to travel faster than c, locally.

I haven't read much about string theory, but I'm speculating that conservation of mass could be said to apply to tachyons, even though they have imaginary mass.

"Stephen Wolfram is the creator of Mathematica and is widely regarded as the most important innovator in scientific and technical computing today." - Stephen Wolfram