The Xamarin.Forms SideDrawer does not provide this functionality. I have logged it as a feature request in our Ideas & Feedback portal. You can track its progress on the following address - SideDrawer: Expose StaysOpen property.
You can "follow the item" in order to subscribe for automatic notifications. I have also added some points to your account for the suggestion.
In the meantime, you can achieve the behavior for the Android platform as the functionality is available there. The following custom renderer should do:
CUstomSideDrawerRenderer : SideDrawerRenderer
.Control.DrawerOpened += Control_DrawerOpened;
sender, Com.Telerik.Android.Primitives.Widget.Sidedrawer.DrawerOpenedEventArgs e)
Have a great rest of the week.
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