Let $\mathbb{C} \ni z \mapsto M(z)$ be a square matrix depending holomorphically on a parameter $z$ with the property that $\operatorname{dim}\ker(M(z)))=1$ for $z $ away from a discrete set $D \subset \mathbb{C}$ and $\operatorname{dim}\ker(M(z)))\ge 1$ for $z \in D.$

I ask: Is it always possible to choose a continuous vector $\mathbb{C} \ni z \mapsto v(z)$ such that $M(z)v(z)=0?$