Use ActionSheet instead of Picker on iOS. https://facebook.github.io/react-native/docs/actionsheetios.html
As answered by jevakallio this is the default behaviour on iOS. But this doesn't give a good UX so remove all picker components and replace with ActionSheet.
I did and it works great. The reason I prefered ActionSheet over other components suggested by jevakallio because it is developed by the RN team and has a good native feeling. The last option suggested react-native-modal-picker is also very good.