Edit: After half day, my wife’s Mac started having the same problem. I have then applied the solution that I found in the Apple
How to Back Up Your Device With iCloud
Whether you have an iPhone, iPad or iPod your iOs device will almost certainly contain important data, such as contacts, documents and applications. Unfortunately even the most finely tuned device can break and there are also plenty of people out there who will not rise above stealing your precious Apple product if you let your guard down, which is why you should always back up the contents of your device, should one of these happen to it. Backing up can be done through iTunes or iCloud and is very easy to do, so if you haven’t done it yet, do it now!
Here is how to do it through iCloud:
iCloud is the backup method that Apple themselves have provided and as such is the ideal place to store your Apple software and other files stored on your device. It is initially free to use, although if you wish to exceed 5GB of space you will need to pay for it, but you can fit a lot into that free amount. If you need more it will cost $20 a year for an additional 10GB.
The first thing you need to do is open Settings, select iCloud and then choose the Storage & Backup option.
You will then see a switch for iCloud Backup which you will need to turn on.
Back on the previous screen you can choose from which apps you wish to back up data. Only the Apple apps already installed on your device will be listed however but you will be able to make a more in-depth selection.
To select further apps, go to Storage & Backup and select Manage Storage.
In the Backups section choose the device you are backing up and wait for the Info page to open up.
On this page you will see a top 5 list of your apps under a heading that says Backup Options but ignore this and select the Show All Apps option instead.
You will then see a complete list of your apps and you can switch the relevant ones on or off to suit your storage needs. Any you turn off will present a red Turn Off & Delete button, which you need to click. This is completely harmless as it only deletes the iCloud backup for that app.
Once this is all complete, connect your device to a Wi-Fi signal and make sure it is plugged in so that battery does not run down during the process and leave the screen locked.
Under these conditions your device will back itself up automatically on a daily basis, although it can also be done manually by going into your Settings, choosing iCloud and then Storage & Backup, and finally selecting Back Up Now.
When the time comes to restore your device, it will present a Restore from iCloud Backup option during the setup procedure.
Selecting this will prompt a request for your iCloud username and password and then you will be able to choose from the three most recent backups. Everything else takes care of itself, although you might need to enter the occasional password when prompted.