A parcel bomb exploded at an apartment building in Monaco, officials said, seriously injuring three people, including a child, and setting off a cross-border hunt for the perpetrator.

Stéphane Thibault, Monaco’s prosecutor, told journalists on Tuesday that a lone man had arrived at the building and dropped off a package on Monday night. Shortly after, three occupants of a ground-floor apartment approached the door, he added.

“The package exploded when those three people arrived,” Mr. Thibault said, adding, “We also have two other bystanders who were injured by projections of glass from storefront windows that exploded a little farther down the street.”

In an interview with the French broadcaster BFM TV, Christophe Mirmand, Monaco’s head of government, said that the reason for the attack was unclear. He added, “No incident of this nature has ever occurred in the principality.”

Monaco, which is smaller than Central Park in New York, is a glamorous tourism destination on the Mediterranean coast. It is known as a tax haven and also hosts a famed annual Formula 1 race.

The three seriously injured victims were a man, a woman and a child, Mr. Thibault, said. As of Tuesday morning, he added, the man was no longer in life-threatening condition, but the woman still was. The child, he said, underwent an operation overnight and was never in life-threatening condition.

Mr. Mirmand and Mr. Thibault declined to identify those wounded.

The Monaco government said in a statement that a suspect had been filmed by surveillance cameras fleeing toward French territory. Mr. Mirmand, in the BFM TV interview, said that the person appeared to have reached Beausoleil, a French town across the border. The French Interior Ministry said that it was working with the Monaco authorities to find the person.

Mr. Mirmand said, “It is very easy to get from the Principality of Monaco to the town of Beausoleil — you just have to cross a street — so at this time it would be difficult to tell you exactly where the person in question is located.”

An investigation was opened into attempted murder and the placement of explosive substances or devices in a public space.

Outlets in France, Monaco and Ukraine said one of the victims was a Ukrainian businessman. The New York Times could not verify that assertion, and the prosecutors did not confirm it.

Prince Albert II, Monaco’s head of state, said in a statement, “The criminal explosion that occurred this evening in Monaco has come as a shock to the entire Monegasque community,” using an adjective that refers to inhabitants of the principality.

“Now more than ever, the Principality of Monaco will remain united and determined in the face of violence and crime,” he added. “The safety of our community has always been a priority; it will remain so, regardless of the threats.”

Andrew E. Kramer contributed reporting from Kyiv, Ukraine.