  We complete the classification of globally generated vector bundles with
small $c_1$ on projective spaces by treating the case $c_1 = 5$ on
$\mathbb{P}^n$, $n \geq 4$ (the case $c_1 \leq 3$ has been considered by Sierra
and Ugaglia, while the cases $c_1 = 4$ on any projective space and $c_1 = 5$ on
$\mathbb{P}^2$ and $\mathbb{P}^3$ have been studied in two of our previous
papers). It turns out that there are very few indecomposable bundles of this
kind: besides some obvious examples there are, roughly speaking, only the
(first twist of the) rank 5 vector bundle which is the middle term of the monad
defining the Horrocks bundle of rank 3 on $\mathbb{P}^5$, and its restriction
to $\mathbb{P}^4$. We recall, in an appendix, from our preprint
[arXiv:1805.11336], the main results allowing the classification of globally
generated vector bundles with $c_1 = 5$ on $\mathbb{P}^3$. Since there are many
such bundles, a large part of the main body of the paper is occupied with the
proof of the fact that, except for the simplest ones, they do not extend to
$\mathbb{P}^4$ as globally generated vector bundles.
