a wild-goose chase
a situation where you waste time looking for something that you are not going to find, either because that thing does not exist or because you have been given wrong information about it.
After two hours spent wandering in the snow, I realised we were on a wild-goose chase.
When I found out that there was no Anita Hill at the university, I began to suspect that I had been sent on a wild-goose chase.