We suggest the $su(1,N|M)$ superconformal mechanics formulated in terms of phase superspace given by the noncompact analogue of complex projective superspace. We parametrized this phase space by the specific coordinates allowing us to interpret it as a higher-dimensional superanalogue of the Lobachevsky plane parametrized by lower half-plane (Klein model). Then we introduced the canonical coordinates corresponding to the known separation of the ``radial'' and ``angular'' parts of (super)conformal mechanics. Relating the ``angular'' coordinates with action-angle variables, we demonstrated that the proposed scheme allows us to construct the $su(1,N|M)$ supeconformal extensions of wide class of superintegrable systems. We also proposed the superintegrable oscillator- and Coulomb-like systems with a $su(1,N|M)$ dynamical superalgebra and found that oscillatorlike systems admit deformed $\mathcal{N}=2M$ Poincar\'e supersymmetry, in contrast with Coulomb-like ones.