Suppose that $M$ is a square matrix with all elements on its main diagonal equal to $1$, and every row containing exactly two off-diagonal elements equal to $-1$; all other elements are equal to $0$. The kernel of $M$ is nonzero and, indeed, contains a vector with all its coordinates nonzero. Does it follow that the row space of $M$ contains a (nonzero) zero-one vector?

In case it matters, the sum of all elements in every column of $M$ is nonpositive, and $M_{ij}M_{ji}=0$ whenever $i\ne j$.