I need to write some details here. But for now, take a look at other's attempts to build single transistor latches:
https://hackaday.io/project/163912-single-transistor-latch
https://hackaday.io/project/7975-one-transistor-latch
https://hackaday.io/project/112126-one-transistor-flipflop
In one of the comments, the reverse breakdown transistor flashed circuit was mentioned. There is a writeup on hackaday about it:
https://hackaday.com/2016/10/17/a-vintage-single-transistor-led-blinker/
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.
This just hit the front page, so be sure to check there for comments - readers may have some insight. https://hackaday.com/2019/05/08/help-solve-the-single-transistor-latch-mystery/