I found this sentence in the short story Two Legs for the Two of Us by James Jones. The larger context is: "George was standing in the door (...) He opened the screen (...) He grinned down at her belligerently through the dark circles and loose lips of an extended bat." What could the word "bat" mean here? None of the usual definitions listed in dictionaries seem to fit this context.
I think this is a baseball metaphor.
bat = a turn at batting (Merriam-Webster)
The author is using extended bat as a euphemism for a long bout of heavy drinking, the effects of which Sandy can see in George's face.