Save often and make two or three manual saves, don't just rely on auto-saves and the quick save trigger option. I made it through the second mission without killing a single person, spent a good amount of time avoiding everyone or setting traps and suchlike yet I got to the end with a single kill on the stats screen.
There is no way to restart the mission and I didn't have a save to go back to so I thought I'd write this here to warn people. I've seen people on reddit say similar things, might be a bug or something but it's damn frustrating. Luckily I was only a few hours in but some people were right at the end only to find they had had a kill and didn't know where it was from.
You can check your kills in the Stats screen on the pause menu, it shows both the current mission and the overall kill count across the game. Check that regularly and save often!
Edit: PuckJunkie added these helpful tips for doing Missions 5 and 8. Refer to them if you're having issues with those particular missions.
*** Spoiler - click to reveal ***There appear to be multiple places in the game where a scripted event results in the deaths of people you've knocked out. I know of two such places (below, in spoiler tags). If this is happening to you, reload a save from before their death occurs, backtrack to them, and shoot their unconscious bodies with a sleep dart. For whatever reason, they still die, but the deaths no longer show up on your statistics.
Mission 5:
*** Spoiler - click to reveal ***The first two witches you come across are outside the main building. If you knock them out, they'll die when you eliminate the target at the end. It's best just to leave them wandering around, but if you've knocked them out, backtrack to them and shoot them again with sleep darts.
Mission 8:
*** Spoiler - click to reveal ***The two smugglers you come across at the very beginning (the "workers" down by the boat) will die as soon as you trigger the conversation between the black market shopkeeper and the woman from the boat. You cannot prevent this - I actually was carrying one on my shoulder when trying multiple approaches, and he still died - but if you shoot them again with sleep darts, their deaths do not show up in your stats.
If anyone else notices events like these then please describe them in the comments and I'll add them in to the solution