Why a spaceship cannot reach the speed of light from the perspective of the spaceship's rest frame

A standard explanation in physics textbooks of why a spaceship, or any material body, cannot reach the speed of light in vacuum is that the increase with velocity of the spaceship’s inertial mass is unlimited as light speed is approached, requiring infinite kinetic energy to reach the speed of light. However, this explanation is provided from the perspective of an inertial frame of reference moving with a constant (sub-light) speed with respect to the spaceship. What stops the spaceship from attaining light speed from the perspective of the spaceship itself, given that the spaceship’s mass is constant in its rest frame? A suitable response is provided to this question.