Are there any gambits which yield an advantage for the side down material? For this question I'm interested only in gambits that:


Are real gambits. That is, the side that's risking the material actually ends up down material + cannot count on winning the material back. This, e.g., excludes the Queen's Gambit because 1. d4 d5 2. c4 dxc4 3. Qa4+ wins the material back without problems (plus most Queen's Gambit lines don't end with White down material). Also, if the gambit cannot be safely accepted, it's excluded.
Don't rely on opponent making a mistake. So the Fried Liver Attack 1. e4 e5 2. Nf3 Nc6 3. Bc4 Nf6 4. Ng5 d5 5. exd5 Nxd5?! 6. Nxf7 doesn't count since 5...Nxd5 is a well-known bad move.


I notice that Stockfish doesn't seem to like gambits. Analyzing using the engine at www.chess.com/analysis, for example:


King's Gambit: 1. e4 e5 2. f4 yields a starting eval of -1.10 (at depth 19).
Evan's Gambit: 1. e4 e5 2. Nf3 Nc6 3. Bc4 Bc5 4. b4 yields a starting eval of -0.60 (d =19).
Danish Gambit: 1. e4 e5 2. d4 exd4 3. c3 yields a starting eval of -0.48.
Sicilian Wing Gambit: 1. e4 c5 2. b4 yields a starting eval of -0.44.
Smith Morra Gambit: 1. e4 c5 2. d4 cxd4 3. c3 yields a starting eval of -0.19
Blackmar-Diemer Gambit: 1. d4 d5 2. Nc3 Nf6 3. e4 dxe4 4. f3 yields a starting eval of -0.49.


(I'd consider Black gambits as well but those are harder to evaluate since the starting position favors White, and therefore it's not surprising if White has an advantage after Black plays the gambit)

Are there any gambits in which the side that is down material for tempo is advantaged?
While it's a gambit by Black, what about Tal's gambit?  Black is scoring 55% in 228 grandmaster and elite correspondence games after 1 e4 c5; 2 f4 d5; 3 exd5 Nf6.  I'd call Black doing better than 50% in that many top games an advantage.  (And 55% is better than Black's score in any of the main non-gambit responses to 2 f4.)