The purpose of this paper is to consider a generalisation of the big Picard theorem of Fujimoto's type for any holomorphic map \(f : \Delta^* \to P^2 \backslash A\), where \(\Delta^*\) is the punctured disk and \(A\) is a curve in \(P^2\) with 4 or more irreducible components in general position in a certain sense and for any meromorphic map \(f : N \backslash B \to P^2 \backslash A\), where \(N\) is an arbitrary manifold, \(B\) is a proper analytic subset of \(N\).