Most likely, your script (and WWW::Mechanize::Firefox are waiting for the new page to load in response to ->click being called. Most likely, you want to look at the synchronize parameter to ->click. Maybe you can help me improve the documentation. Currently it describes synchronize as

synchronize - Synchronize the click (default is 1)

Synchronizing means that WWW::Mechanize::Firefox will wait until one of the events listed in events is fired. You want to switch it off when there will be no HTTP response or DOM event fired, for example for clicks that only modify the DOM. ...

Also, WWW::Mechanize::Firefox::Troubleshooting mentions Clicking on a link makes the Perl script wait forever, which also matches your problem description. What changes in the documentation could have made you find this better?