Proton Movies The app is designed specially for playing audio books.

You have to manually download the books and place them in subfolders under the "My Audiobooks" folder on your phone.
Every book must be in a separate subfolder even if it consists of only one file.
In Library→Settings→Root folder select the Audiobooks folder.
Once finished don't forget to press the "Update" button at the top of the Library window.

First 30 days Full version. Later - Basic version.
Features:
+ Playback speed control.
+ Classification of books (new started finished).
+ Download cover from the Internet.
+ Bookmarks.
+ List of characters. You can manually create a list of characters to easier follow the story.
+ Automatic pause in case you fall asleep. To continue playback just shake your phone.
+ Playback history.
+ Chromecast support.
+ Application Widget. Allows you to control the player from the home screen.
+ You don't have to finish one book to start another. The progress is saved independently for all books.
+ NO ADS!

To purchase full version press: menu--Help--Version tab.
Full version cost is $2 + tax.

Many thanks to people who left comments and suggestions.
If you have something that doesn't work please write an email instead of leaving a comment.

Version for Android 4.4 - 5.1:
https://drive.google.com/file/d/159WJmKi_t9vx8er0lzTGtQTfB7Aagw2o

Version for Android 4.1 - 4.3:
https://drive.google.com/file/d/1QtMJF64iQQcybkUTndicuSOoHbpUUS-f/view?usp=sharing

Version with old icon:
https://drive.google.com/open?id=1lDjGmqhgSB3qFsLR7oCxweHjnOLLERRZ