Let $R$ be the ring $K[X]/(X^r)$ for some field $K$ and $r>1$, and consider the module $XR$. I can show that this module is not projective by observing that the ring is local, so all projective modules are direct sums of copies of $R$, and $XR$ is not one of these because it doesn't have the right dimension.
In particular, this means that $XR$ cannot satisfy the lifting property, but I haven't been able to find an example of a map that doesn't lift - can anybody think of a good example in this case?
Looking around previous questions suggests that if the module I want to show isn't projective is a quotient of the ring then the identity generally won't lift, but there doesn't seem to be a neat strategy for dealing with ideals.