A lithium battery with a solid electrolyte is still a lithium battery.
A true solid state battery will use a silicon anode (and most likely a solid electrolyte). The benefit being silicon will hold 10X the charge of the current graphite of the same size, and cost less, and take a charge essentially like a capacitor - all at once.
My guess is it won't be a car company that figures this out. The ramifications are way beyond vehicles. Imagine enough grid storage that you could run completely on green or renewable energy, a cell phone or laptop that legitimately would run a week without charging, etc.
https://spectrum.ieee.org/silicon-anode-battery