An iPhone or iPad does not allow the storing of audio files unless they were purchased from the iTunes store.
However, there is a way around this conundrum and that is to Download and save to your Dropbox by creating a free account, or sign in to an existing account.
These audiobooks can be played as a single file in Dropbox (choose play offline) but as they are in audiobook format they are best listened to in this wonderful free app, BookPlayer, available at the iTunes store. Click icon link to get the app:
Ideally get the app BookPlayer before you download the book.
For a more enjoyable experience, if you have installed Book Player, open the app
and click: + Add , import files,
locations and choose Dropbox,
then click on the audiobook file and wait till it has finished importing.
You can now begin listening with a lovely interface and with all the enhanced features of Audible.