This game mode would likely be ideal in 3v3 or 4v4. The idea is that one player from each team is the "V.I.P." and only V.I.P. kills count.
V.I.P. could be visually differentiated (Light effect, banner, or paint job perhaps?), but wouldn't necessarily need any gameplay modification. i.e. They are a regular ship than can shoot and repair as normal. Thus they can choose to be one of the attackers attempting to kill the other team's V.I.P. or they can just run and hide. (Risk / Reward)
Games could last until one team's V.I.P. is killed, or perhaps killed a certain number of times.
Selecting the V.I.P. could be at the lobby screen (i.e. first slot on a team is the V.I.P. slot). This would allow teams to place their most (or least?) experienced player as the V.I.P. and the others as defenders/attackers.
Alternately, V.I.P. could be randomly assigned when the match starts.
Optional Consideration 1) the V.I.P. could have a perma-spot or map marker (so that there is no true hiding). The other team always has an indication of where you are. If this is TOO easy, the V.I.P. indicator could be rougher and identify location by map grid or compass quadrant (i.e. NE/NW/SE/SW).
Optional Consideration 2) the V.I.P. could have defensive bonuses (more armor, reduced rebuild times, etc). This would make playing as the V.I.P. easier, but would be more challenging to implement and balance. I don't think bonuses would be necessary for this game mode.