If you get an inbox message from a user regarding FREE GIFT CARDS OR SIMILAR, the account does not appear to be a real staff account and the site is probably phishing. Please click the report button and do not use the link as it could reveal your Fiverr account information to a scammer.
EDIT: Fiverr management is aware of the issue and that multiple scam messages are being sent. They are doing all they can to stop it. Please don’t post about it with the URL as someone might get confused and click your link. Just ignore them and wait it out. I will re-open this thread so that people can reply here if they have questions or comments instead of starting new threads.
In addition, if you are worried about response rate, there is no need to panic. You can reply to the messages with anything (like “scam”) if you wish and then re-report it. You may not be able to catch them all, though, and you don’t really have to do that. If you contact customer support very nicely one time after this is over, ask them if they can reset your response rate due to a recent series of scam emails. Make it simple and nice. They will help.