You can always ask. But Apple doesn't specifically "refurbish" iPhones. They might offer repairs or a service replacement. In general the requirement is that an equivalent device must be exchanged, which is why Apple says that repair service may be restricted to the country of purchase. There are some unlocked iPhones that are meant for distribution in both the US and Canada. The LL region code in the model number generally means the US but can also be distributed in Canada. But if it was ever carrier locked then you'll probably need to go to the US for service.
What's the model number in Settings > General > About > Model Number ?