The Hopf link is a simple example of a link with two components that is not an unlink.
The Borromean rings form a link with three components that is not an unlink; however, any two of the rings considered on their own do form a two-component unlink.
^ abKanenobu, Taizo (1986), "Hyperbolic links with Brunnian properties", Journal of the Mathematical Society of Japan, 38 (2): 295–308, doi:10.2969/jmsj/03820295, MR0833204