Book files can only do save as

In addition to @Eugene Tyson 's sage advice, techincally, it is the individual documents that are updated when saving book changes such as sync'ing and updating numbers. 

Try this until you get it worked out. 

Open all the documents by selecting all of them in the book file and double-clicking on any one of them.

Update your numbering.

Save all the open documents at once by pressing Cntl-Alt-Shift-S (Win) or Cmd-Opt-Shift-S (Mac) at once.

You can close them all by pressing Cntl-Alt-Shift-W (Win) or Cmd-Opt-Shift-W (Mac) at once.

Now, you can reset your preferences using the InDesign Preference dialog box. Do that, quit, and restart.

Backup your Book file - as a copy somewhere safe

Then delete your book file in the current folder (keep your backup safe for restoring later if needed)

File>New Book and re-add your files

Save your book file

Does this resolve  your issue? 

It could be file permissions or syncing issues.

Sync or cloud folders

If the book file is in OneDrive, Dropbox, Google Drive, or another synced folder, InDesign may force a “Save As” because the file is temporarily locked. Moving the file to a local folder can prevent this.

Permissions

Make sure you have full read/write access to the folder containing the book file.

Network drives

Working on network locations can cause similar issues. Copying the file locally while editing can help.

Preferences corruption

Resetting InDesign preferences can resolve odd behaviours, though it resets all settings.