Bizarre. That didn't happen in any of my tests and I'm not getting replication from start.
Does the Choose/Name Investigator screen show the investigator as starting with negative stam?
If not, what's the listed starting stats for the investigator and when you get in-game what stats do they have at the start of the game? (The one with the single box on the right side that's been struck as not implemented and is talking about the fog)
e: Oh, and though it's not pointed out you can use the Go Home button to return to your holding and do Short Rests to gain 1d3 stam and 1d3 will at the expense of a 'terror step'.
e2: Aha, I've found it. Looks like it doesn't successfully 'draw a new Investigator' when you've already died once. Also failed to discriminate between stats when checking for last-hit death between stam and will. Though it's not a perfect remedy, closing and re-opening the game should fix it for the time being. Thank you very much for the bug report, I'll see both of these are fixed in a patch I'll be building today.