In this paper, we introduce codes equipped with pomset block metric. A Singleton type bound for pomset block codes is obtained. Code achieving the Singleton bound, called a maximum distance separable code (for short, MDS ($\mathbb{P},\pi$)-code) is also investigated. We extend the concept of $I$-perfect codes and $r$-perfect codes to pomset block metric. The relation between $I$-perfect codes and MDS $(\mathbb{P},\pi)$-codes is also considered. When all blocks have the same dimension, we prove the duality theorem for codes and study the weight distribution of MDS pomset block codes when the pomset is a chain.