The Pirates of the Caribbean actor, 43, and his partner Vanessa Paradis, 34, are holding a vigil at the child's bedside after she was struck down by the undisclosed illness nine days ago.
The Mirror claims it knows what medical condition Lily-Rose is suffering from, but have decided to withhold it from the press out of respect for the privacy of Depp's family.
A hospital source says, "He has not asked for any special favors but just wants the best for his child, like any parent.
"It is such a difficult time for him but he is hanging in there for the sake of his daughter, and the whole family is just praying that she gets better. They are taking each day as it comes.
"It's been an emotional rollercoaster for all of them and he's been worrying himself silly. He's a devoted dad and is spending as much time as he can with Lily."
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