Is whether there is a complete negatively curved surface embedded in the unit ball still an open problem? I was reading the text "Open problems in geometry of curves and surfaces" by Mohammad Ghomi, and on page 15 Problem 7.2 asks "are there any complete negatively curved surfaces embedded in the unit ball?"
I tried googling to search for solutions to this problem, but was only able to find general definitions of complete negatively curved surfaces unrelated to the problem in this question.
Is this still an open problem?
 A: The latest update of Ghomi's list of open problems  is from September of the 2019 and the problem is still listed as open. If it was solved in the last 6 months, Ghomi probably would have updated his survey (and, most likely, the word of a solution would be out). Thus, you can safely assume that the problem is currently open. (I would not recommend to work on this problem though, unless you are an established mathematician who can afford to spend time tackling a well-known open problem, or if you are comparable, in the sense of mathematical strength and maturity, to John Pardon, when he was an undergrad in Princeton.) 
One more thing: If you check Mathoverflow discussion here, you might get an impression from Tom Mrowka's answer that Hadamard's problem was solved by Colding and Minicozzi in 2008. This is not the case. What they proved is that a complete embedded minimal surface of negative curvature in $E^3$ must be unbounded. But Hadamard's problem was about arbitrary surfaces, not minimal ones. One can say that they proved that Nadirashvili-type examples  cannot be embedded.  
