You should replace <i>$object</i> with <i>$:item.object</i> and ensure that the attributename for the binary file in the class is file. If not, replace <i>file</i> with the attributename.
Congratulations! Would you mind explaining how you solved this and what code you used. For at least one person (me) is looking for a similar solution, but the information above is not sufficient for me.
We put the files (pdfs in our case) in one folder (node id 296). With the following code we can display the pdfs with a direct download link in a template.
{* Fetch everything that is under node #296 (children, grand-children, etc.) *}
{let nodes=fetch( 'content', 'tree', hash( 'parent_node_id', 296 ) ) }
{* Loop through the nodes and display their names. *}
{section loop=$nodes}
<br />
<span class="fett">{$:item.data_map.titel.data_text}</span>
<br />
{$:item.data_map.beschreibung.data_text}
<br />
<img src={"pdf_mini_icon.png"|ezimage}
<a href={concat("content/download/",
$:item.object.id,
"/",
$:item.object.data_map.file.id,
"/file/",
$:item.object.data_map.file.content.original_filename)|ezurl}>
{$:item.object.name|wash}</a>
<br />
{/section}
{/let}