Suppose $M$ is a c.t.m and suppose $P$ is $Fn(I,2)$ where $I$ is infinite. Now suppose $G$ is $P$-generic, and $A \in M[G]$ is infinite set.

Is it guaranteed that the exist $B \in M$ such that $B \subset A$, and $B$ is infinite ? If not, is there any 'reasonable' condition that we can add to $M$ or $P$ to get the desired result ? (Like CH is $M$, or large enough $I$)