Summary: For an element \(\Psi\) in the graded vector space \(\Omega^\ast (M,TM)\) of tangent bundle valued forms on a smooth manifold \(M\), a \(\Psi \)-submanifold is defined as a submanifold \(N\) of \(M\) such that \(\Psi_{\vert N} \in \Omega^\ast (N,TN)\). The class of \(\Psi \)-submanifolds encompasses calibrated submanifolds, complex submanifolds and all Lie subgroups in compact Lie groups. The graded vector space \(\Omega^\ast (M,TM)\) carries a natural graded Lie algebra structure, given by the Frölicher-Nijenhuis bracket \([-,-]^{FN}\). When \(\Psi\) is an odd degree element with \([\Psi,\Psi]^{FN}=0\), we associate to a \(\Psi \)-submanifold \(N\) a strongly homotopy Lie algebra, which governs the formal and (under certain assumptions) smooth deformations of \(N\) as a \(\Psi \)-submanifold, and we show that under certain assumptions these deformations form an analytic variety. As an application we revisit formal and smooth deformation theory of complex closed submanifolds and of \(\varphi \)-calibrated closed submanifolds, where \(\varphi\) is a parallel form in a real analytic Riemannian manifold.