I solved it. I saw my mistake. Height="110"
This was one of the reason why list didn't dropdown.
The class is correct.
Constructor should look like this:
List<cBank> oListBanks = new List<cBank>();
oListBanks.Add(new cBank {Name="Alior Bank", first_session_in = ...});
oListBanks.Add(new cBank {Name="Bank BPH",first_session_in=...});
...
this.ListPicker.ItemsSource = oListBanks;
And XAML code should be:
<toolkit:ListPicker x:Name="ListPicker" Header="Bank:" >
<toolkit:ListPicker.ItemTemplate>
<DataTemplate>
<StackPanel Orientation="Horizontal">
<TextBlock Text="{Binding Name}" FontSize="30" HorizontalAlignment="Center"/>
</StackPanel>
</DataTemplate>
</toolkit:ListPicker.ItemTemplate>
<toolkit:ListPicker.FullModeItemTemplate>
<DataTemplate>
<StackPanel Orientation="Horizontal">
<TextBlock Text="{Binding Name}" Width="150"/>
</StackPanel>
</DataTemplate>
</toolkit:ListPicker.FullModeItemTemplate>
</toolkit:ListPicker>
Now all works properly.