I see where i got this wrong. As already mentioned by Kent, the line
Template="{StaticResource PushpinControlTemplateP}"
shouldn't be in MapItemsControl.
I solved my problem with moving this line of code to Resources so they look like this:
<UserControl.Resources>
<ControlTemplate x:Key="PushpinControlTemplateP">
<Grid>
<Ellipse Fill="Green" Width="15" Height="15" />
</Grid>
</ControlTemplate>
<DataTemplate x:Key="PushpinDataTemplateP">
<m:Pushpin Location = "{Binding MapLocationP}" ToolTip="{Binding MapTooltipTextP}" Template="{StaticResource PushpinControlTemplateP}"/>
</DataTemplate>
</UserControl.Resources>