If only thermodynamics was this easy. Heat pumps ain't magic, alas, they are still subject to physical limitations, including the laws of thermodynamics.
Also keep in mind that cooling the hot part of your space heat pump is very limited: there's no convection nor conduction in space. You can only lose heat energy via radiation or ablation (= shooting away hot pieces).
I'm talking about waste heat from when you use the electricity.
Almost no matter what machine you are powering, be it a a toaster or a computer or an electric car or a washing machine, eventually turns all of the electric energy into heat.
(You can contrive some counter-examples. Eg if you point a sufficiently strong laser pointer at the sky, some of the energy will escape earth before turning into heat here.)
Even if you can turn 100% of the energy you received from space into electricity, in the end it'll all turn to heat.