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.
Mars
Joseph Eoff
YSPACE Labs
Dr. Cockroach
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/