When TextBox IsReadOnly property is true, no need to show caret.

samples/ControlCatalog/Pages/TextBoxPage.xaml

@@ -9,6 +9,7 @@
               Gap="16">
       <StackPanel Orientation="Vertical" Gap="8">
         <TextBox Text="Lorem ipsum dolor sit amet, consectetur adipiscing elit." Width="200" />
+        <TextBox Watermark="ReadOnly" IsReadOnly="True"/>
         <TextBox Width="200" Watermark="Watermark" />
         <TextBox Width="200"
                  Watermark="Floating Watermark"

src/Avalonia.Controls/TextBox.cs

@@ -282,7 +282,10 @@ namespace Avalonia.Controls
             }
             else
             {
-                _presenter?.ShowCaret();
+                if (!IsReadOnly)
+                    _presenter?.ShowCaret();
+                else
+                    _presenter?.HideCaret();
             }
 
             e.Handled = true;