There are rechargeable lithium ion batteries (CR-123), but unprotected (non current limited) cells are very dangerous when used in multi-cell applications, if one of the batteries is old, or defective, leaving the fresh cell to lead the pack and deliver dangerous current which may result in an explosion.
I've used rechargeable lithium-ion cells for RC, imported flashlights (Chinese, which use the same cells used in laptops), and they are fantastic for the energy they deliver, but have to be recharged with care (balanced, in multiple cell applications). They become unstable if discharged beyond a certain point as well, and the same applies to overcharging. Smart chargers are a must!
Just do a search on CR123 rechargeable batteries. I only use primary cell CR123s, for most of my weapon light applications, because the lights are very rarely used.
Again, please find protected rechargeable CR123 batteries, if they are available, if that is what you want.
You might want to do a search on rechargeable lithium ion explosions and fires, to get a feel for the danger and risk involved.