The attributes responsible for rendering the current file path to an instantiated form us found in: forms.widgets.ClearableFileInput()
Override ClearableFileInput()'s render method. In my case I changed
substitutions['initial'] = (u'<a href="%s">%s</a>'
% (escape(value.url),
escape(force_unicode(value))))
to
substitutions['initial'] = (u'%s <br />'
%(escape(force_unicode(os.path.basename(value.url)))))
Doing this removed the path to the current file and displays just the file name, without a link.