Floating plane in PCB
You can simply extend your ground regions out towards the edge of the board.
If you are not designing for extremely low cost at extremely high volume, you can simply leave those areas blank (no copper). Most board shops won't charge noticeably extra for large etched areas in prototype quantities.
You can use copper thieving to fill those areas. This is numerous small disconnected copper areas. It is good for manufacturability because it ensures copper is balanced between the different layers and reduces the rate that etchant is depleted when etching the boards, but at the same time it is not likely to affect your circuit because relatively the copper regions are relatively small with relatively large gaps between them and active areas of the circuit (so there's little capacitance to couple signals).