Is there a model of ${\sf ZF}$ such that there is an infinite set $X$ and a injective map $f:{\cal P}(X)\to {\cal P}(X)$ so that for $a\neq b \in {\cal P}(X)$ we have $|f(a)\cap f(b)| \leq 1$?

Is there a model of ${\sf ZF}$ such that there is an infinite set $X$ and a injective map $f:{\cal P}(X)\to {\cal P}(X)$ so that for $a\neq b \in {\cal P}(X)$ we have $|f(a)\cap f(b)| \leq 1$?

Note. As user Gro-Tsen in the comments below points out, if we weaken the condition $|f(a)\cap f(b)| \leq 1$ to "$f(a)\cap f(b)$ is finite", then the resulting statement is a theorem of ${\sf ZFC}$.