Do PHP static variables persist across Batch API batch items processing? Do PHP static variables, which do persist during a normal page request, persist between progressive Batch API runs? What about during non-progressive (command-line) batch execution?  Does &amp;drupal_static come into play in any way?

In my application, I want to take advantage of in-memory caching across batch processes.
Static variables exist per request, so no, they won't persist across separate batch runs.

If you need to cache data across requests in memory just use redis/memcache/etc, there are contrib modules available for each that hook into the core caching stays so you can use a familiar API