As weidehopf points out, there is a lot of confusion created by the casual use of the term “lithium battery”.
Most consumer devices including vape pens (I’m not a user - I had to look that up) use rechargeable Lithium Ion batteries. There is no metallic lithium in them and the recommended method for extinguishing fires in these batteries is to use water, preferably by immersing them in water.
Lithium batteries on the other hand are not rechargeable and DO contain metallic lithium. The most common lithium batteries are button batteries. There are also larger ones - for example my SPOT GPS tracker uses AA disposable lithium batteries.
Metallic lithium reacts violently with water, so dousing a fire with water would not be a good idea! A Class D dry powder extinguisher is required to extinguish a lithium battery fire.