Hi,
You can add a check for that exact site and based on that show/hide search web part. Click on the blue arrow next to 'Visible' checkbox on the web part and add macro:
{% SiteContext.CurrentSiteName != "DancingGoat" |(identity)GlobalAdministrator%}
This will return false only for your site and search web part won't be visible.
Best regards,
Dragoljub