LONDON—The U.K. government has suffered its first setback in its bid to introduce legislation to allow the country to begin the process of withdrawing from the European Union, but Prime Minister Theresa May said her plans to start exit talks by the end of March would remain on track regardless.

The short bill, which would empower Mrs. May to notify the EU of its intention to leave, was overwhelmingly approved by the House of Commons last month. But on Wednesday members of Parliament’s unelected upper chamber, the House of Lords, voted to amend the legislation to include a guarantee for the roughly 3 million EU citizens living in the U.K. to stay after Brexit by a margin of 358 to 256.

The amendment means the House of Commons will have to reconsider the bill, potentially setting off a “ping pong” game which could delay its passage as the legislation is batted back and forth between the two chambers until a compromise is reached. Both houses must approve legislation for it to be passed into law.

In a statement, the government’s Department for Exiting the EU said it was “disappointed” the House of Lords had chosen to amend a bill that the House of Commons had passed without amendment. It reiterated that the government wanted to guarantee the rights of EU citizens in Britain and Britons in the EU as soon as possible.

The government was forced to put forward the bill to trigger Article 50, the formal mechanism for leaving the EU, after the country’s Supreme Court ruled it couldn’t proceed without parliamentary approval. Though the defeat for the government in the House of Lords is unlikely to be any more than a speed bump, it is a fresh sign of the continued skepticism of some lawmakers toward Brexit, and comes after two former prime ministers who campaigned to stay in the EU criticized the government’s approach.

“It is indeed my plan to trigger [Article 50] by the end of March,” Mrs. May told Parliament earlier when asked about the looming vote in the House of Lords and recent comments by former Conservative Prime Minister John Major and former Labour Prime Minister Tony Blair.

The defeat in the House of Lords came despite government efforts to reassure peers on the eve of the vote that Britain and the EU had a common goal to offer reciprocal assurances on citizens’ rights as quickly as possible.

During the debate, Dianne Hayter, a peer from the opposition Labour Party, argued that EU citizens in Britain were too precious to be used as bargaining chips to secure the rights of U.K. nationals living in the EU.

“These two groups of people should not be traded against each other,” she said, adding that areas of the economy relied heavily on EU employees, such as the state health service and construction.

Sarah Ann Ludford, a peer from the opposition Liberal Democrat party, said, “If the U.K. government gives a unilateral guarantee to the millions of EU citizens contributing in this country not only will it do the right thing morally and economically but it will also supply the crucial catalyst for a quick reciprocal deal.”

But Conservative peer Michael Howard questioned the point of the amendment and noted that the House of Commons had already debated the question of rights for EU citizens and the government wasn’t going to change its mind.

“There are no new facts in this debate,” he said, adding that the best way to end the uncertainty about the rights of EU nationals was to pass the bill as quickly as possible and activate Article 50.

Mrs. May wants to control immigration from the EU, removing Britain from Europe’s tariff-free single market for trade and establishing a separate free-trade agreement with the bloc. In January, she said she had told EU leaders that she was ready for an agreement on reciprocal rights for EU citizens in Britain and Britons in the EU but some countries had blocked an early deal. EU nations have said they won’t negotiate with the U.K. until it has triggered the Article 50 process.

The defeat for the government came after Mr. Major on Monday said Britons were being offered an “unreal and overoptimistic” vision of Brexit and said the costs of leaving the EU would be substantial. His successor, Mr. Blair, earlier said it was his mission to persuade Britons that they should have the right to change their minds about Brexit when the eventual terms become clear.