The 19-year-old became the first player born since the turn of the millennium to play in Germany's top flight in 2017-2018, but he could not stop Hamburg's relegation.
Arp scored twice in 18 league appearances in that campaign, but he has yet to get off the mark in the second tier as Hamburg lead the way after 20 matches.
He will continue his development at Hamburg this season before a decision is made on his future - he could move to Bayern in 2019 or 12 months later.
"The negotiations with Bayern Munich were constructive and fair," said Hamburg sporting director Ralf Becker.
"Relegation hit Fiete very hard and he wanted to stay here and help us get back up.
"He puts everything into achieving that goal, and nothing will change in the future."
Arp announced himself on the international scene in 2017 when he scored 10 goals in eight games at the UEFA Under-17 Championship, before netting five at the Under-17 World Cup.