In this note we present new examples of determinantal point processes with infinitely many particles. The particles live on the half-lattice ${1,2,\ldots}$ or on the open half-line $(0,+\infty)$. The main result is the computation of the correlation kernels. They have integrable form and are expressed through the Euler gamma function (the lattice case) and the classical Whittaker functions (the continuous case). Our processes are obtained via a limit transition from a model of random strict partitions introduced by Borodin (1997) in connection with the problem of harmonic analysis for projective characters of the infinite symmetric group.