As far as I understand, viewed from Earth's reference frame, a spaceship accelerating by constant force will get heavier and heavier as it gets faster, and thus get less and less speed out of it, asymptotically approaching the speed of light but never exceeding it. Moreover, crucially, any pingbacks from clocks on the spaceship we observe will appear to be progressively going slower than they should. (Time dilation)
The spaceship's own reference frame will be a whole different story. If I understand correctly, it will perceive its own mass to be constant, the rest of the universe's relative speed to asymptotically approach the speed of light, but also the rest of the universe to get progressively squashed in the direction of travel (length contraction dual to time dilation), with the two effects multiplying up in such a fashion that the Newtonian relation between the integral of your acceleration and the time it takes to the destination seems to hold throughout.
In other words, for the spacefarer's schedule, it's as if the speed of light does not actually play a role: they can always accelerate more/go arbitrarily faster to get to their destination faster. However, from their point of view, it is as if the universe around them will deform in the process to accommodate this, and from the stationary observer's point of view, it is as if they are actually still moving slowly but tricking themselves into thinking otherwise by accelerating their passage of time.
The spaceship's own reference frame will be a whole different story. If I understand correctly, it will perceive its own mass to be constant, the rest of the universe's relative speed to asymptotically approach the speed of light, but also the rest of the universe to get progressively squashed in the direction of travel (length contraction dual to time dilation), with the two effects multiplying up in such a fashion that the Newtonian relation between the integral of your acceleration and the time it takes to the destination seems to hold throughout.
In other words, for the spacefarer's schedule, it's as if the speed of light does not actually play a role: they can always accelerate more/go arbitrarily faster to get to their destination faster. However, from their point of view, it is as if the universe around them will deform in the process to accommodate this, and from the stationary observer's point of view, it is as if they are actually still moving slowly but tricking themselves into thinking otherwise by accelerating their passage of time.