As an extension to that:
http://www.apple.com/legal/warranty/products/iphone-english.html
Note this statement from mobile device warranties:
"IMPORTANT RESTRICTION FOR iPHONE, iPAD, APPLE TV, HOMEPOD, AND APPLE VISION PRO SERVICE.
Apple may restrict warranty service for iPhone, iPad, Apple TV, HomePod, and Apple Vision Pro to the country where Apple or its Authorized Distributors originally sold the device."*
The above is from the USA version of this iPhone, iPad and iPod warranty document, but this particular section limiting international coverage is much the same for the warranty for all countries. If you wish to check the warranty for where you bought your device, click on this link to take you to the page about Apple warranties, then select your device and country --> Legal - Hardware Warranties - Apple
If you are not in the country where the device was purchased, there is no guarantee of getting warranty repair. Your best chances are to ask to speak with an Apple supervisor or Apple Store manager (not a lower tier employee) who is familiar with the warranty (i.e., that the “may” means it does not outright deny any kind of service) and who will be trying to work with you to find a satisfactory arrangement. Apple wants you to be a satisfied customer. If you visit a store, make sure it is an Apple Store. An authorized service provider (AASP) may only have limited access to parts and may be mostly interested in selling things. You want an actual Apple employee who is higher up and will be more likely to try to work with you for a solution. Keep in mind you may also have to be prepared to not get an identical configuration of device. iDevices and parts from another region model may not be available in another country. You will ultimately have to decide what kind of compromises you are willing to accept or if it is best to wait until the item can be returned to the country where you bought it. You would have to do that yourself, or arrange for a friend in the country to send it in for repair. Apple does not do international shipping.
We do not know the details of your interaction with the Apple salesperson. They may not have understood you wanting to take the iPhone to Italy, or they may not know the details of the warranty and coverage as applies to other countries.
