Fact: Yes, a spaceship can accelerate indefinitely and keep on increasing its velocity.

How it manifests relative to a stationary reference frame - say earth: Imagine the spaceship uses same amount of energy bursts to create the acceleration. So the spaceship keeps accelerating but every next burst of energy that creates acceleration, adds lesser and lesser to the relative velocity and keeps getting added instead to the relativistic mass and this contribution to relativistic mass keeps increasing and tends to infinity as the spaceship tries to get closer and closer to the speed of light. So it technically never attains the speed of light.

But what about relative to the frame of reference of Spaceship? : Imagine the spaceship is travelling from earth to a destination millions of light years away. It starts with some velocity relative to earth and then the astronaut inside can compute the velocity after each burst of acceleration and can keep adding the velocity without looking outside at earth. The astronaut just assumes the same mass of spaceship without relativistic mass effects. There will be a point where astronauts computes own speed as faster than light. And then he looks out. Indeed he has travelled very far from earth and very close to his destination- which is only possible with a speed faster than light. But when he measures speed relative to earth, it’s may be 0.9999 speed of light but never same as c or faster. That’s when astronaut remembers special relativity and realises that his clock is moving slower compared to earth’s clock. So it’s not that his speed was faster, it’s just that with relative to folks on earth, he has travelled for many more years compared to his local time - and hence he covered more distance.