I know that there are projects that try to map brain connectivity. My question is whether there is a dataset available with the complete connectivity of just a single minicolumn of the neocortex. Doesn't have to be human.
Unfortunately not. The closest publication would be Lee et al 2016, which is a reconstruction of a network of functionally-characterised neurons in mouse V1. They state the data "will be available" at neurodata; currently only the EM dataset appears to be there, and not the correspondence with functional imaging. I encourage you to contact them for the data set.
Alternatively, there are functional measurements and small networks available from mouse visual cortex Cossell et al. 2015 (self-cite; data available on request); and anatomical statistical measurements available from rat somatosensory cortex Markram et al. 2016 (data available from NMC).