It's not super uncommon. I've done it before - mostly when in a contracting position. I've also had direct reports that trusted me enough to give me as a reference - mostly when I'd been encouraging them to jump into career advances that I'd been unable to provide for them where we were.

But yeah, it;s 100% understandable and acceptable to not want to give any current employer references. (Which is a bit tricky for the OP, who's only got relevant professional experience with their current employer...)