Yes, but you also need nearly infinite fuel.

An efficient engine could use antimatter as its battery/fuel source and move itself forward by using antimatter to eject propellant close to the speed of light in the opposite direction right? That probably wouldn’t require a large mass of antimatter/propellant. Especially since you can collect propellant from the interstellar medium wherever you’re going

Even antimatter has limits, and those limits deny reaching a relativistic gamma of 1e6/27:

http://www.projectrho.com/public_html/rocket/enginelist3.php

But even if you ignore that limit, at that kind of speed you also need to account for your outer hull undergoing significant damage as each stray intergalactic hydrogen you hit has a kinetic energy of 34.75 TeV in your frame of reference, and while this will create a large supply of antimatter, it will do so in a way significantly less helpful to your goal than the fact we can transmute lead into gold by using the former as a radiation shield in a nuclear reactor is useful to someone who wants to get rich quick.

Bonus fact: this energy level is higher than the LHC, so you may have some extra weird mass-modification effects besides relativity as the front of your spaceship is now a Higgs Boson torch.

Nah, just 3e18 kg ;)

That's a more difficult problem compared to DNA repair, crew rotation, or material erosion.