Yes, the industry is recycling their batteries at a fairly high rate.
Most of the recycling is post-industrial. After a battery is made, it's tested, and if it fails testing, it is recycled.
There hasn't been a large amount of post-consumer car battery recycling because there aren't many 20 year old car batteries out there. The reuse market for old batteries is higher than expected, and people reusing batteries pay more than recyclers. But the post-industrial plants are humming along, and will expand to accept post-consumer batteries once the supply arrives.
Everybody brings up the plastic-recycling bugbear. There are a bunch of reasons why plastic recycling is hard, most of which don't apply to batteries.
A better example is lead-acid battery recycling. The US has a >99% lead acid battery recycling rate. Lithium batteries are much larger and have a much higher content of valuable materials, so will be much more fruitful to recycle than lead acid batteries.
The main materials recovered out of batteries are nickel, copper and manganese, which are quite valuable. Lithium is secondary -- lithium batteries don't actually contain a lot of lithium, and lithium is cheap and common. New plants are set up to recover lithium because the price of lithium has increased, but old plants don't.