Madrid fell at Real Betis 1-0 for their first defeat since the return of coach Jose Mourinho.
PSG fell at home to Monaco 2-1 on the same day that coach Luis Enrique agreed to stay at the European champion through 2030.
Betis stunned Madrid when newcomer Troy Parrott, an Irish striker, knocked in a ball that Thibaut Courtois did well to save from a header by a Betis teammate.
Madrid best chance to reply wasn't until Vinícius Júnior earned a penalty when he was cut down by Natan in the box in injury time. But Betis goalkeeper Álvaro Vallés smothered Mbappé's spot kick, adding to his great performance on the night.
Madrid had won their first three games in La Liga.
Reigning European champions PSG looked to be on their way to their first win when Marquinhos put the host ahead in the 31st, but Monaco rallied with goals by Flavio Nazinho in the 58th and Stanis Idumbo in the 72nd.
Monaco lead Ligue 1 as the only team to win their first three games.
PSG had already drawn 2-2 with Rennes and Lille in their first two games. They also lost the French Super Cup to Lens 1-0 last month.
Their first league defeat came hours after the French powerhouse announced contract extensions for coach Luis Enrique and several players.
João Neves, Lucas Beraldo, Senny Mayulu and Willian Pacho all got new deals through 2031, while Fabian Ruiz had his contract extended through 2028.
