I assume that you are using control nodes for your objects? If so, This is the behavior of the controls nodes.
Try putting an empty control node (inputblocker) the size of your window at the bottom of your scene tree, but above your unpause stuff as shown in the hierarchy below. Make sure the inputblocker is hidden when the game is not paused. It should mainly just be below other control nodes.
Let me know if this works for you.