Relativistic spin-polarized positron beams are indispensable for future electron-positron colliders to test modern high-energy physics theory with high precision. However, present techniques require very large scale facilities for those experiments. We put forward a novel efficient way for generating ultrarelativistic polarized positron beams employing currently available laser fields. For this purpose the generation of polarized positrons via multiphoton Breit-Wheeler pair production and the associated spin dynamics in single-shot interaction of an ultraintense laser pulse with an ultrarelativistic electron beam is investigated in the quantum radiation-dominated regime. A specifically tailored small ellipticity of the laser field is shown to promote splitting of the polarized particles along the minor axis of laser polarization into two oppositely polarized beams. In spite of radiative de-polarization, a dense positron beam with up to about 90% polarization can be generated in tens of femtoseconds. The method may eventually usher high-energy physics studies into smaller-scale laser laboratories.