  We propose a novel semi-discrete Kadomtsev--Petviashvili equation with two
discrete and one continuous independent variables, which is integrable in the
sense of having the standard and adjoint Lax pairs, from the direct
linearisation framework. By performing reductions on the semi-discrete
Kadomtsev--Petviashvili equation, new semi-discrete versions of the
Drinfel'd--Sokolov hierarchies associated with Kac--Moody Lie algebras
$A_r^{(1)}$, $A_{2r}^{(2)}$, $C_r^{(1)}$ and $D_{r+1}^{(2)}$ are successfully
constructed. A Lax pair involving the fraction of $\mathbb{Z}_\mathcal{N}$
graded matrices is also found for each of the semi-discrete Drinfel'd--Sokolov
equations. Furthermore, the direct linearisation construction guarantees the
existence of exact solutions of all the semi-discrete equations discussed in
the paper, providing another insight into their integrability in addition to
the analysis of Lax pairs.
