I have encountered the following property. Can anybody tell me if it already exists in literature and/or is equivalent/similar to other well-known properties?

Property:$(X,d)$ metric space. For any open ball $B\subseteq X$ and for any distinct $x,y\in B$, there exist two disjoint open balls $B_1\ni x$ and $B_2\ni y$ and two open continuous and injective functions $f_i:B\rightarrow X$ such that $f_i(B)\subseteq B_i$.

Well.. it's similar to contractibility, but seems to be *weaker* - it's some density condition..