The answer, in a word, is… both. Marble Hill is the neighborhood which makes up the northernmost part of the Borough of Manhattan but because of the re-routing of the Harlem River it is physically located in the Bronx. According to wikipedia:
“In 1895, the construction of the Harlem River Ship Channel rendered the Marble Hill an island bounded by the canal to the south and the original course of the Harlem River to the north. The Greater New York Charter of 1897 designated Marble Hill as part of the Borough of Manhattan. Effective on January 1, 1914, by an act of the New York State Legislature Bronx County was created, but Marble Hill remained as part of New York County. Later in 1914, the old river was filled in, physically connecting Marble Hill to the borough of The Bronx and the rest of the North American mainland.”
So geographically, Marble Hill does not lie on the island of Manhattan, but rather in the Bronx. And, interestingly from an administrative perspective, Marble Hill belongs to both Manhattan and the Bronx. For example while it is part of the Borough of Manhattan, it falls within two Bronx Community Districts and receives Fire and Police Service from within the Bronx.