Seems like there is plenty of lithium, but mines are not keeping up with demand. Here's an interesting story from FT about Chile's lithium mines, with pictures:
It takes time to construct new mines. When Tesla started building the gigafactory, they signed long-term contracts with companies building new mines in the US and Mexico.
https://www.ft.com/content/cde8f984-43c7-11e6-b22f-79eb4891c...
(You may need to Google for the title to bypass paywall: "Lithium: Chile’s buried treasure")