Can they say they have evidence they don't have?

Generally yes, and it's a standard technique rather than an aberration.

Common versions: claiming fingerprints or DNA were recovered, claiming surveillance footage shows you, claiming a co-defendant has given a statement implicating you, claiming a witness picked you out. Any of these may be entirely fabricated.

The purpose isn't to inform you. It's to create the impression that denial is futile so that explaining becomes the only apparent option — and an explanation is a statement.

This tactic is particularly effective on innocent people, who often reason that a confused detail must be their own memory failing rather than the officer inventing evidence. That reasoning has produced false confessions in documented cases.

What to do with it: nothing. There's no way to test the claim in the room, and engaging with it means talking. If the evidence exists, it will appear in discovery and can be examined then, with counsel, on paper.