The well known fact about $\ell^1$ says that the Schauder basis of $\ell^1(I)$ behaves more-less like a Hamel basis, namely if $X$ is any Banach space and $\mathcal{E}=(e_i)_{i\in I}$ is a basis for $\ell^1(I)$ then any bounded map $f\colon \mathcal{E}\to X$ can be extended to a bounded operator. Say that $\ell^1(I)$ is projective in that case.
Let me change the setting to nuclear spaces.
Is it true that if $Z$ is a nuclear Montel space with a basis $\mathcal{E}$ then any bounded map $f\colon \mathcal{E}\to Z$ can be extended to a bounded (thus continuous, since the space is neccessarily metrizable) operator?
In particular, I am interested mainly in the space $Z=H(\mathbb{D})$ of holomorphic functions on the disc topologized by the usual family of supnorms on the smaller closed discs.