Asklemmy

It depends on what you mean by "escape", and what you view as the alternative.

I suspect that the pursuer could never converge on the same instantaneous point, given sufficient initial distance (and orientation). At a certain distance, the prey could enter a stable orbit around the pursuer. I don't have a mathematical proof but I strongly suspect this to be the case,and I can envision the structure of a proof.

Could the prey infinitely extend the gap between themselves and the pursuer? No. I don't have the tooling to actually present such a proof, but of that one I am confident.

I think if you introduced concepts of obstacles and a "radius of escape" (where if the gap meets a threshold the predator is permanently foiled), then there are almost certainly scenarios where the prey could escape.

We actually see this scenario play out in nature all the time

It can never escape because its turning speed helps nothing while the distance is big, so the pursuing ship can always catch up to it again.
The only reason a fighter pilot has a chance to escape a faster missile is when the missile's targeting system can only see in front of it, so when it overshoots it loses its target.
But with a faster turning speed, the chased ship can evade the pursuer forever, if the captain always turns at the perfect moment.

Any amount of turn in between 0 and 180 i presume, and the game is "tag" correct? Do they have to be caught from the back, or would a head to head collision count? Does the trailing ship have perfect response time? I don't see how turning around 180 degrees would help evade capture. In summary, no, I think you'll always be tagged by the faster ship.