Metric spaces not isometric to any of their proper subsets

Question

Let's say a metric space $X$ has property $P$ if $X$ is not isometric to any of its proper subsets. I'd like to know what this property is called in the literature and whether there's a nice characterization of spaces having this property.

What I know so far:

1. Compactness implies property P, so compactness is a sufficient condition,
2. The set $ \mathbb{Z}$ of integers ( with Euclidean metric) is not compact but satisfies $P$, so compactness is not necessary.

Any necessary or sufficient conditions ( weaker than compactness) are also appreciated.

Thanks in advance.