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>
