I don't think we can call that time dilation, as it doesn't actually involve anything to do with time itself
Strictly speaking, actual time dilation probably doesn't either.
If true time dilation were going on, the time on the ship would slow down, so that the crew would only perceive, for example, five days in transit instead of five months
Actually, this is only true when measured from an observer in a remote reference frame. Say, if you were on Earth and you had a telescope pointed at the window of a ship that was moving past you at 90% the speed of light. In that case, YOU would see that time on the other ship has slowed down considerably relative to your own measurements. The crewman on that ship, however, look out the window with their own telescope and they observe the exact thing about you: from their perspective, YOU'RE the one moving at 90% the speed of light, and therefore YOU'RE the one who's experience time dilation.
Both observations are equally valid. They would HAVE to be, after all, or else a photon emitted from your location wouldn't appear to be traveling at lightspeed in both reference frames, it would be faster in one than the other. Relativity tells us this can NEVER happen, which ironically means that time dilation is only present when the relative velocities REMAIN high; if the fast-moving vessel were to suddenly stop, the dilation vanishes altogether.
A better way to illustrate this is to have three or more ships traveling at different speeds to the same destination. One is moving at 90% of C, one at 92%, one at 88%. The middle one observes two spacecraft moving away from it at 2% of the speed of light and gets identical time dilation measurements from both of them. The two on the ends, however, see one ship moving away at 2% and the other moving away at 4%, and the faster of the two has a greater degree of time dilation than the slower. All three measurements will be entirely contradictory, and all three measurements will be correct. But since their OWN internal clocks appear to be unaffected, they will eventually land on a nice planetoid somewhere, put their watches on the table and find out that actually NONE of them have experienced any time dilation at all. They will, of course, look back at their point of origin and think "Well we were moving at nearly lightspeed, so back home, it should only be a few minutes after we left." They will be wrong, of course, but if they whip out a telescope and look, that's EXACTLY what they'll see.
The moral of the story is that relativity can seem to be a very strange thing, until you realize the broader implication of the universe: Just because you see it doesn't mean it really happened.