Binding для элемента Picker в Xamarin.Forms

Используйте this.BindingContext, как в примере : Bindable Picker Control

C#:

public class RegistrationPageViewModel : INotifyPropertyChanged
{
    ....
    List<string> countries = new List<string>
    {
        "Afghanistan",
        "Albania",
        "Algeria",
        "Andorra",
        "Angola",
        ...
    };
    public List<string> Countries => countries;
    ...
}
public partial class RegistrationPage : ContentPage
{
    RegistrationPageViewModel vm;
    public RegistrationPage()
    {
        InitializeComponent();
        this.BindingContext = vm = new RegistrationPageViewModel();
    }
}

XAML:

<Picker ItemsSource="{Binding Countries}" />