I need to write some details here. But for now, take a look at other's attempts to build single transistor latches:
In one of the comments, the reverse breakdown transistor flashed circuit was mentioned. There is a writeup on hackaday about it:
Someone suggested using the negative resistance region as a latch, similar to neon tube latches. I don't think anyone attempted it, otherwise we would have heard about it. I tried to find a voltage where the transistor would either stay on or stay off, based on its previous state, and couldn't find it. So I started experimented with different biasings. Also added and removed capacitors to try to slow down the response of the circuit so I could watch what was happening. Then I found it: Add a resistor between the base and the collector. It works. But I'm not sure why.