What you need is a DECT headset, Jabra makes several models, as does Yealink and HP.

HP sells the headsets under the Poly and Savi brands, these are the same headset series that Plantronics used to make, I remember how weird is was that Plantronics was first bought by Polycom who became Poly, only to almost immediately after that got bought by HP.

I miss Plantronics, HP have gutted the series, I miss my CS540, and later a modern variant that worked directly with a computer and didn’t need a special freak cable. But my hearing has got worse and I like my stereo Jabra headset at work.

I am actually considering buying a Jabra Engage 75 SE, though a 45 SE would be brilliant as well.