I’m curious about the possibility of being contacted regarding what could be a fake job.

I saw a job posting on LinkedIn. I was contacted almost immediately when I applied - first red flag, to me anyway. I tried to find a website at which this person supposedly worked, but one doesn’t exist. I was going to ignore it, but just to confirm my suspicions, I responded telling her I couldn’t find the website. She claims they are a small recruitment company and their website is under construction - second red flag. she then said the position is part time and a contract role, which I don’t want, so I told her it wasn’t what I was looking for. Today, she contacted me about another role, contract to perm, full time. I don’t know. Something just doesn’t sit well with me. Has anyone here had any experience being contacted about a possible fake job? I'm going to stick with my gut, but wondered how prevalent these are. Thanks