cold fish

An aloof person.

A hard-hearted, unfeeling individual, one who shows no emotion, as in Not even the eulogy moved him; he's a real cold fish. This expression was used by Shakespeare in The Winter's Tale (4:4): "It was thought she was a woman, and was turn'd into a cold fish." However, it came into wider use only in the first half of the 1900s.

The noun has one meaning:

Meaning #1: an aloof unemotional person