A very simple wrapper around IMGUI's AdvancedDropdown for UI Toolkit.

AdvancedDropdownField.cs

public class AdvancedDropdownField : DropdownField
{
    #region Types

    private class SimpleAdvancedDropdown : AdvancedDropdown
    {
        #region Fields

        private readonly DropdownField _dropdownField;

        private readonly string _title;

        private readonly List<string> _choices;

        #endregion

        #region Constructors

        public SimpleAdvancedDropdown(DropdownField dropdownField, string title, List<string> choices) : base(
            new AdvancedDropdownState())
        {
            _dropdownField = dropdownField;
            _title = title;
            _choices = choices;
        }

        #endregion

        #region Methods

        /// <inheritdoc />
        protected override void ItemSelected(AdvancedDropdownItem item)
        {
            _dropdownField.index = item.id;
            _dropdownField.tooltip = item.name;
        }

        /// <inheritdoc />
        protected override AdvancedDropdownItem BuildRoot()
        {
            var root = new AdvancedDropdownItem(_title)
            {
                id = -1
            };
            for (var i = 0; i < _choices.Count; i++)
            {
                var choice = _choices[i];
                root.AddChild(
                    new AdvancedDropdownItem(choice)
                    {
                        id = i
                    });
            }

            return root;
        }

        #endregion
    }

    #endregion

    #region Fields

    private readonly SimpleAdvancedDropdown _advancedDropdown;

    #endregion

    #region Constructors

    public AdvancedDropdownField(string dropdownTitle, List<string> choices, int index) : base(dropdownTitle, choices, index)
    {
        _advancedDropdown = new SimpleAdvancedDropdown(this, dropdownTitle, choices);
        tooltip = choices[index];
        RegisterCallback<PointerDownEvent>(OnPointerDown, TrickleDown.TrickleDown);
        AddToClassList(alignedFieldUssClassName);
    }

    #endregion

    #region Methods

    private void OnPointerDown(PointerDownEvent evt)
    {
        var popup = this[1];
        if (evt.button != 0 || !popup.worldBound.Contains(evt.position))
        {
            return;
        }

        evt.StopImmediatePropagation();
        _advancedDropdown.Show(new Rect(evt.position, Vector2.zero));
    }

    #endregion
}