According to Portuguese media reports, the British baby and child were rescued by passersby. The tram crashed at around 6pm local time in the Lapa area of the country’s capital. Several people were taken away on stretchers, and transported to Santa Maria and San Francisco hospitals.
But firefighters said they didn’t believe any of the injuries suffered were serious.
Pictures of the devastating scene show the tram appearing to be completely flipped over.
It is seen lying in a dark street surrounded by debris as emergency services rush to the scene.
A witness told Portguese online newspaper Publico the tram had failed to stop and appeared to be in disrray after a failed attempt to brake.
It then derailed at the intersection between Rua Garcia de Orta and Rua de São Domingos before crashing into a building.
The witness added he heard “a bang” and “people screaming”.
The Civil Protection said more than 50 rescuers had been deployed locally, including the Ourique and Lisbon Firefighters.