Set x:null as the style to be the default style.
If you are using MaterialDesignThemes, you can overwrite the resource for the key named 'MaterialDesignDatePickerTextBox'.
<StackPanel>
<StackPanel.Resources>
<ResourceDictionary Source="pack://application:,,,/MaterialDesignThemes.Wpf;component/Themes/MaterialDesignTheme.Defaults.xaml" />
</StackPanel.Resources>
<Border Margin="10">
<DatePicker Style="{x:Null}"><!-- Clear Style -->
<DatePicker.Resources>
<Style TargetType="{x:Type DatePickerTextBox}">
<Setter Property="Text" Value="{Binding RelativeSource={RelativeSource Mode=FindAncestor,AncestorType=DatePicker},Path=SelectedDate,StringFormat=dd/MM/yyyy}" />
</Style>
</DatePicker.Resources>
</DatePicker>
</Border>
<Border BorderBrush="Black" BorderThickness="1" Margin="10">
<DatePicker >
<DatePicker.Resources>
<!-- Inherit style of MaterialDesign -->
<Style TargetType="{x:Type DatePickerTextBox}"
x:Key="MaterialDesignDatePickerTextBox"
BasedOn="{StaticResource MaterialDesignDatePickerTextBox}">
<Setter Property="Text" Value="{Binding RelativeSource={RelativeSource Mode=FindAncestor,AncestorType=DatePicker},Path=SelectedDate,StringFormat=dd/MM/yyyy}" />
</Style>
</DatePicker.Resources>
</DatePicker>
</Border>
</StackPanel>