The minimum basis set consists of the smallest number of functions necessary to contain the electrons of the neutral system. In the book Introduction to Computational Chemistry, Second Edition, by Frank Jensen, it is stated that "a set of p-functions" usually is added to the minimum basis sets of Li and Be atoms, despite them containing 3 and 4 electrons respectively and thus would require 2 s-functions. I can not, however, understand why.
By your definition, any minimal basis set would only include as many orbitals (including spin state) as there are electrons. This implies that each atom would be described as a single determinant and therefore the entire system would be described by a single determinant. Without unoccupied orbitals, there is no binding. Hence, a minimum basis set has to fulfill a few more constraints. A common requirement is the inclusion of the full valence shell and invariance under spin-flip. For second period elements, this includes p-orbitals.