The non-Fourier heat transfer in a half-space is analyzed under sudden heating or cooling on a local surface. The non-Fourier heat transfer effect is described by the time-fractional dual-phase-lag (DPL) model, where the fractional derivative without singular kernel is used. An axisymmetric mixed initial-boundary value problem is solved by the use of the Hankel and Laplace transforms. Two typical cases of sudden temperature rising on a circular zone of the surface or an instantaneous surface heat source are analyzed. For sudden temperature rises, the heat flux and temperature gradient exhibit an inverse square-root singularity near the boundary of the heating zone and their dynamic intensity factors are computed numerically in the time domain. For the instantaneous surface point heat source, an exact solution of the transient temperature at any position in the Laplace domain is obtained. The effects of the fractional order and relaxation time on the temperature distribution and heat flux response are elucidated. The singular behavior of the transient thermal response and the non-Fourier effect of heat transfer are shown.