How to add a "Script" field in custom inspector?

In the default Inspector there’s field referencing a script file. Is there any way to do the same in custom inspector?

You can use the EditorGUILayout.ObjectField function :

MonoScript script = null;
script = EditorGUILayout.ObjectField(script, typeof(MonoScript), false) as MonoScript;

This script will allow you to select any existing script in your project in this field.