Depends on constraints.

How far is their separation distance?

How fast is the turning, and how fast are the ships? If it takes a year to turn around, and a day to "escape," then yes.

What kind of ship? Under-sea, sea, air, space? And what size?

What does the field look like? Open, hazards, obstacles, maze?

Conditions of the medium? Stormy, Choppy, Calm?

How near is "caught" and how far away is "escape?" Is "caught" within range of long distance weaponry? Is escape a destination or a condition?

It depends, but let's assume a simple scenario of a sea ship in calm open ocean, with a turning radius of 5 ship lengths, a ship length of 40 meters, and a speed of 37.04 km/h.

• Weaponry included and coming broadside is the bad end for the slower ship: the slower ship doesn't stand a chance.
• Boarding required: It'll be a long game of cat and mouse, but the cat will win. However, if they were of equal speed it'd likely be an infinite game.

Even assuming open 2D sea and a pursuer turn radius of, let's say, 20 ship lengths, I'm sure that depends on boundary conditions like how far it has to go to escape and the position/speed/orientation the ships start with.

Obviously, if we start with a pursuer right off the stern, there's no escaping.

For real ships, there also will be a time limit, because someone will run out of supplies first. The way I imagine doing this would basically be to find a pattern where no matter what the pursuer does they can't board, but it doesn't matter if you run out of hard tack while the people chasing are still well-fed. Cannons would change the logic here too.

Guess:

The slower ship would never be in a position to get caught without making a mistake, but it's turning advantage rapidly reduces with with distance, so it would never be able to get away in the long run.

Confounding factors: Ship lengths. As the ships get longer relatively their turning circle, the easier the slow ship becomes to catch. The length of the ship makes it harder to dodge.

Conditions of capture and escape. Are we talking about being side by side, close enough to through grappling hooks, or just a point touching? Or cannon range? Is escaping getting out of line of sight? RADAR range?

Islands. This favours the more manoeuvrable ship as the fast ship cannot take a straight line course. It also increases the fast ships cost for falling for bluffs.

Acceleration and speed cost to turn. Overall, introducing these makes the engagement longer. That said, it would also affect the fast ship more if islands are introduced.

Outside objectives. Ships can't be at sea forever, and likely both ships have places to be. Is a vital shipment of vaccines on the slow ship? Does the fast ship know this? Etc

Ill defined. Slower/faster than what? The pirate ship? Whats the initial distance between the ships? I'll assume a sufficiently large distance so the ships all get away :)