I know from experience that when I check out a Kindle book from the library and download it to my Paperwhite that as long as I do not sync my Paperwhite, I can keep and read that library book for as long as I want.
Using that model, I downloaded this book onto my old Nexus 5, which I don't use as a phone but keep as a backup. I started listening to the book and, as my brother thought, I love it. I was halfway done last night and knew it expired at midnight. So I put the Nexus 5 in airplane mode.
This morning I checked and wahwah, no joy. When I clicked on it, it told me the book was expired. Sigh.
Happily, in this case, I was able to recheck the book out right away and now I have another 21 days. Whew.
This book is a fictional memoir of a woman exactly my age. She (the character) chronicles the times that I lived and it's great fun to remember with her. And now I can finish it.