  For every nuclear $\mathbb Z_\ell$-algebra $\Lambda$ and every small v-stack
$X$ we construct an $\infty$-category $\mathcal D_{\mathrm{nuc}}(X,\Lambda)$ of
nuclear $\Lambda$-modules on $X$. We then construct a full 6-functor formalism
for these sheaves, generalizing the \'etale 6-functor formalism for $\Lambda =
\mathbb F_\ell$. Prominent choices for $\Lambda$ are $\mathbb Z_\ell$, $\mathbb
Q_\ell$ and $\bar{\mathbb Q_\ell}$ and especially in the latter two cases, no
satisfying 6-functor formalism has been found before. Applied to classifying
stacks we obtain a theory of nuclear representations, i.e. continuous
representations on filtered colimits of Banach spaces.
