While debugger tooltips has some interactivity handy, they are still based on tooltip window and behave accordingly, therefore you cannot make it persistent. This is by design, debugger tooltips are supposed to allow quick examination of variable value and no more else.
When you need to examine a variable more closely and you are actually interested in the deeper details like private members, you could examine your structured variable in the debug inspector window. (one per variable). You can open many such windows, they are persistent during debugging session and can be docked to your debugging layout. Note that having
button pressed continuously retrieves property values with getter method.

For debugging aids which are persistent across the debugging sessions and are less detailed see @SilverWarior's answer.