//Your data structure is fine. The issue is that you're not creating multiple option elements, you're only creating one. This part is off:
$("#cityPicker").append('<option value="' + value
.districts + '">' + value.districts + '</option>');
}
//what you want to do is create an option element for each district, as follows:
for (const district of value.districts) {
$("#cityPicker").append('<option value="' + district + '">' + district + '</option>');
}