In most instances, the clock does stop and you get to resume your search with whatever time is remaining from when you halted the search. But in other cases, it’s not quite that simple.
You can’t jump on a ride-on mower and take out the fescue on the right side of the second hole in an effort to find your ball. But if you happen to accidentally rip some grass out as you reach down into the rough and scratch around for your ball, there is no penalty according to Rule 7.1.