Another way to think is to consider two space walking astronauts; one inside the ship and one outside. Neither is touching the ship, both are moving at essentially the same speed in the same direction. All three pretty much stay together.
However, there could be a teeny tiny amount of acceleration experienced by each. For example, at an extremely high velocity, even the tiny impulse caused by each interstellar proton hitting a body can cause a bit of drag. The "indoor" space walker won't experience it, and so won't be slowed at all, but the ship will, and so will the "outdoor" space walker. It's not clear which one would be affected more, it depends on their cross-sectional areas and masses.
Then there are tidal effects. If there is a distant gravitational source, and there always is, that will accelerate all three the same. But if you are fairly close to a source of gravity, then it is possible that it affects them slightly differently because they will each have a very slightly different distance from the source.
For more on that see answers to Lowest ISS microgravity and for fun see How to get sunburned through the window of a General Products hull?
And before your ship does another neutron-star flyby to accelerate so fast, remember that what humans call UV is not the only thing that gets through a General Products Hull!