I think the answer is no, as shown by the following set $A$ (whose elements I will describe as subsets rather than binary functions).
For any map $\sigma:\omega\to\omega+1$ \sigma:\omega\to\omega+1:=\omega\cup\{\omega\}$ define its hypograph $$\mathrm{hypo}(\sigma):=\{(m,k)\in\omega\times\omega: k < \sigma(m)\}\subset\omega\times\omega\ .$$
An increasing map $\sigma:\omega\to\omega+1$ takes the value $\omega$ if and only if its hypograph contains a subset of the form $[n,\omega)\times \omega$, thus, if and only if it has the property (P) stated in the question (there exist infinitely many $m$ such that there exist infinitely many $k$ such that $(m,k)\in \mathrm{hypo}(\sigma)$; which is indeed verified quite in a strong sense)way).
Let $A$ be the set of all hypographs of all increasing maps that take the value $\omega$. Clearly, the set $A$ is countable, with no isolated points; and all its elements enjoy property (P). A point in $\mathrm{cl}(A)\setminus A$ is exactly the hypograph of an increasing $\omega$-valued map, which never satisfy satisfies property (P).
