We present an experimental study of time refraction of spin waves (SWs) propagating in microscopic waveguides under the influence of time-varying magnetic fields. Using space- and time-resolved Brillouin light scattering microscopy, we demonstrate that the broken translational symmetry along the time coordinate results in a loss of energy conservation for SWs and thus allows for a broadband and controllable shift of the SW frequency. With an integrated design of SW waveguide and microscopic current line for the generation of strong, nanosecond-long, magnetic field pulses, a conversion efficiency up to 39% of the carrier SW frequency is achieved, significantly larger compared to photonic systems. Given the strength of the magnetic field pulses and its strong impact on the SW dispersion relation, the effect of time refraction can be quantified on a length scale comparable to the SW wavelength. Furthermore, we utilize time refraction to excite SW bursts with pulse durations in the nanosecond range and a frequency shift depending on the pulse polarity.