If the faster ship is smart I can't see a way for them to fail to catch the nimble ship eventually.

I think if the faster ship is "dumb" and always just drives straight at the slower ship, then the slow ship might be able to keep itself moving perpendicular to the fast ship which will be unable to spend it's entire speed advantage catching up since it can't keep itself pointed at slow ship.

There's a distance component here, since the closer the ships are to each other the more significant difference in turning speed becomes. (At long distance, one unit of turn might equal ten units of distance, but at close distance one unit of turn might only equal one unit of distance. In math terms, as the radius gets smaller, so does arc length)

I don't care to do the math at the moment, but at some distance between them, the distance the slow ship can travel in a time unit will be greater than the arc length the fast ship can turn at that distance in that time.
If the slow ships turning rate is high enough to keep it perpendicular at that distance, then it should be able to avoid capture, but never escape. (My gut says turning radius equal to or smaller than the distance between the ships, but I could be wrong).
If the ship can't turn fast enough to make a circle, then it'll grow further away than the magic distance and the fast ship will be able to point at them and get closer. If the faster ship is smart, it can just increase the distance and make a more favorable approach by pointing where the slow ship will be.