Holomorphy of a function with values in a Hilbert space

No. There are non-holomorphic functions $f:\mathbb D\to \ell^2$ such that all components $f_n=\pi_n\circ f$ are holomorphic. This follows from a general result of Arendt and Nikolski [Vector-valued holomorphic functions revisited. Math. Z. 234 (2000), no. 4, 777–805]:

Theorem 1.5 Let $X$ be a Banach space and $W$ a subspace of $X'$ which does not determine boundedness. Then there exists a function $f : \mathbb D \to X$ which is not holomorphic such that $\varphi \circ f$ is holomorphic for all $\varphi\in W$.

Here, a subspace $W\subseteq X'$ is said to determine boundedness if, for all subsets $B\subseteq X$, the condition $\sup\lbrace |\varphi(x)|: x\in B\rbrace <\infty$ for all $\varphi\in W$ implies that $B$ is bounded. This theorem is applied to $X=\ell^2$ and the linear span $W$ of all projections $\pi_n$. It does not determine boundedness by considering $B=\lbrace ne_n:n\in\mathbb N\rbrace$ with the unit vectors $e_n\in\ell^2$.