The showcase provides an example for DropDownMenu where you have a search option at http://...richfaces/component-sample.jsf?demo=dropDownMenu However, when you click into the search input field the DropDownMenu disappears, and no input is possible.
The source code directly taken from the showcase:
<
rich:dropDownMenu
label
=
"Search"
direction
=
"bottom-left"
jointPoint
=
"br"
styleClass
=
"search"
mode
=
"client"
>
<
rich:menuItem
>
<
h:inputText
value
=
""
id
=
"searchinput"
/>
<
button
type
=
"button"
>Search</
button
>
</
rich:menuItem
>