Goetz, 21, who joined the club as a teenager in 2008, was plucked out of the reserves this term and established himself in the defence.
"Ashton proved with his Bundesliga and Under-23 performances that he has massive potential," Hamburg sports director Peter Knaebel said on Tuesday. "It is this potential that we want to further develop."
Hamburg, who are two points off the bottom in 14th place, have released Macedonian winger Valmir Nafiu to join Apoel Nicosia in Cyprus on a deal to 2020.
(Reporting by Karolos Grohmann, editing by Tony Jimenez)
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