Thank you for your answer, Ken! Now it WORKS but I did it a little bit differently.

<a4j:mediaOutput element="a" cacheable="false" createContent="#{calendarController.exportCalendar}"

                            value="#{event.calendar}" mimeType="text/Calendar">

                            <h:outputText value="#{event.calendar.name}" />

</a4j:mediaOutput>

public void exportCalendar(OutputStream os, Object calendar) {

     ... // handle calendar  to  outputStr

     FacesContext fc = FacesContext.getCurrentInstance();

     Object response = fc.getExternalContext().getResponse();

     ...

     ResourceResponse rresp = (ResourceResponse) response;

     rresp.reset();

     rresp.setContentType("text/Calendar");

     rresp.setProperty("Content-disposition", "attachment; filename=\"" + filename + "\"");

     ...

     try {

           os.write(outputStr.getBytes(Charset.forName("UTF-8")));

          os.flush();

     } finally {

          os.close();    

     }

    ...

}

I did not use command fc.responseComplete();, is it ok?

2nd possible solution:

When I tried the way similar to http://stackoverflow.com/questions/10451095/downloading-a-csv-file-using-jsf again with "<h:commandLink ....><f:ajax /></h:commandLink>",

I have now ResourceResponse but  file was not returned to user. My code was very similar to the code in this URL, but instead of HTTPServletResponse I used object of type ResourceResponse. Do you know why it does not work? There is no exception after running my code...

But the first solution works! I am just interested in why the second solution does not work...

Thx!

Ok, thank you very much, Ken!