Proceeding from a nonlinear realization of the most general N = 4, d = 1 superconformal symmetry, associated with the supergroup D(2, 1; α), we construct a new model of nonrelativistic N = 4 superconformal mechanics. In the bosonic sector it combines the worldline dilaton with the fields parametrizing the R-symmetry coset S² ∼ SU(2)/U(1). We present invariant off-shell N = 4 and N = 2 superfield actions for this system and show the existence of an independent N = 4 superconformal invariant which extends the dilaton potential. The extended supersymmetry requires this potential to be accompanied by a d = 1 WZW term on S². We study the classical dynamics of the bosonic action and the geometry of its sigma-model part. It turns out that the relevant target space is a cone over S² for any non-zero α±1/2. The constructed model is expected to be related to the `relativistic' N = 4 mechanics of the AdS₂ × S² superparticle via a nonlinear transformation of the fields and the time variable.