  The Fornax galaxy cluster is the richest nearby (D ~ 20 Mpc) galaxy
association in the southern sky. As such, it provides a wealth of oportunities
to elucidate on the processes where environment holds a key role in
transforming galaxies. Although it has been the focus of many studies, Fornax
has never been explored with contiguous homogeneous wide-field imaging in 12
photometric narrow- and broad-bands like those provided by the Southern
Photometric Local Universe Survey (S-PLUS). In this paper we present the S-PLUS
Fornax Project (S+FP) that aims to comprehensively analyse the galaxy content
of the Fornax cluster using S-PLUS. Our data set consists of 106 S-PLUS
wide-field frames (FoV ~ 1.4 x 1.4 deg$^2$) observed in five SDSS-like ugriz
broad-bands and seven narrow-bands covering specific spectroscopic features
like [OII], CaII H+K, H$\delta$, G-band, Mg b triplet, H$\alpha$, and the CaII
triplet. Based on S-PLUS specific automated photometry, aimed at correctly
detecting Fornax galaxies and globular clusters in S-PLUS images, our dataset
provides the community with catalogues containing homogeneous 12-band
photometry for ~ 3 x 10$^6$ resolved and unresolved objects within a region
extending over ~ 208 deg$^2$ (~ 5 Rvir in RA) around Fornax' central galaxy,
NGC 1399. We further explore the EAGLE and IllustrisTNG cosmological
simulations to identify 45 Fornax-like clusters and generate mock images on all
12 S-PLUS bands of these structures down to galaxies with M$\star \geq 10^8$
M$\odot$. The S+FP dataset we put forward in this first paper of a series will
enable a variety of studies some of which are briefly presented.
