The morality of space colonisation is yet to receive thorough examination from philosophers. I seek to address this deficit by making a case against colonising space. This case rests on a defence of antinatalism through four arguments (an argument from Immanuel Kant’s categorical imperative; David Benatar’s axiological asymmetry argument; Benatar’s argument from quality of life; and Benatar’s misanthropic argument) and an accompanying stance in favour of (non-objectionable) extinction. I respond to challenges and objections against both positions and show that they fail. Because space colonisation, if successful, will likely extend humanity’s lifespan, this makes it a morally indefensible activity, insofar as it entails suffering on the part of sentient beings. I consequently argue that space colonisation ought to be prohibited. I recommend that this prohibition take the form of various additions and changes to existing international space law