Theresa May was warned by the French president, François Hollande, at their first meeting in Paris that the UK cannot expect access to the single market if it wants to put immigration controls on EU citizens.

At a joint press conference in the Élysée Palace, Hollande made it clear that the new British prime minister was facing a choice about whether to accept free movement of people in return for free trade.

Standing next to May and speaking in French with an official translator, he said: “It’s the most crucial point. That’s the point that will be the subject of the negotiation.

“The UK today has access to the single market because it respects the four freedoms. If it wishes to remain within the single market it is its decision to know how far and how it will have to abide by the four freedoms.

“None can be separated from the other. There cannot be freedom of movement of goods, free movement of capital, free movement of services if there isn’t a free movement of people … It will be a choice facing the UK – remain in the single market and then assume the free movement that goes with it or to have another status.”

Hollande’s comments suggest it will be difficult for the UK to fulfil the desire of Boris Johnson, the foreign secretary, and other prominent leave figures during the referendum campaign, who favoured access to the single market while imposing limits on immigration.

The French president offered more support over May’s decision to wait until next year before triggering article 50 of the Lisbon treaty, which formally kicks off the two-year process of the UK leaving the EU.

Earlier in Dublin, Hollande had demanded an explanation for the delay, saying: “We understood it would be September, then October and then December. Justifications will have to be given.”

However, he appeared to soften his language after the bilateral talks with May, saying he “understood the government that has just been formed needs this time”.

He went on: “But let me repeat, the sooner the better in the common interests of Europe … because uncertainty is the greatest danger. When economic players doubt the conditions under which the UK will leave and the relationship that will be maintained, there can be risks for stability of the European economy and therefore for jobs.”

The two leaders found most consensus on the issue of maintaining the existing Le Touquet agreement that means UK border checks are conducted in Calais in an attempt to control the flow of refugees and migrants across the channel.

During the EU referendum campaign, May, David Cameron and a French government minister all suggested this could be in jeopardy and the border might move to the UK if there was a vote for Brexit. These claims were dismissed by leave campaigners as “project fear”.

On Thursday, May and Hollande said they were completely committed to maintaining the Calais border.

May said: “We have discussed the Le Touquet agreement, and President Hollande and indeed interior minister [Bernard] Cazeneuve have both been very clear from their point of view that they wish the Le Touquet agreement to stay. I want the Le Touquet agreement to stay.

“I know there are those who are calling for it to go. There are those within France who are calling for it to go … Le Touquet is of benefit I believe to both the UK and France and we are both very clear, Britain now having taken the decision to leave the EU, Le Touquet agreement should stay.”

Hollande became the first EU leader to guarantee that British people living in France would be allowed to stay in his country once the UK’s exit negotiations conclude.

“There is no doubt that the French people who reside in the UK will be able to continue to work there and that the British people in France will be able to continue to work there and spend as much time as they wish,” he said.

May has held off making a promise guaranteeing the right to stay of all EU citizens in the UK until she gets pledges from other nations that the rights of Britons will be preserved.

“I expect to be able to do so, and the only situation in which that wouldn’t be possible is if British citizens’ rights in European member states were not being protected,” she said.

May arrived in Paris after her visit to Berlin for similar talks with Angela Merkel. The German chancellor struck a sympathetic note about it being right and necessary for Britain to take its time with preparations for triggering article 50. Unlike Hollande, Merkel did not rule out a deal that combines some level of access to the single market with controls on immigration.

In the coming months, May is expected to visit more EU leaders as she lays the foundations for negotiations on Brexit, even though Brussels has banned formal and informal talks until article 50 is activated.

Cameron used his final meeting with EU leaders in Brussels earlier this month to warn that the British public would be unwilling to accept any deal that did not include limits on the free movement of people.

But there are concerns among other member states that ceding ground to Britain on the issue of immigration controls – which became a central theme of the referendum campaign – would strengthen the hand of anti-immigrant parties in other countries. Marine Le Pen, the leader of France’s far right Front National, was jubilant after the Brexit vote, calling it the most important moment since the fall of the Berlin Wall.

May has handed the tough task of negotiating the details of the EU departure to David Davis, who will run Whitehall’s new Brexit ministry, while two more Brexiters, Liam Fox and Johnson, have been put in charge of trade and foreign affairs, both key departments as the practical challenges of negotiating an exit emerge.

Brexit campaigners appeared to suggest during the hard-fought referendum campaign that Britain would be able to maintain tariff-free access to EU markets while also “taking back control” of migration flows, but refused to identify what specific type of relationship they had in mind.

Cameron fought hard for the right to control migration during his renegotiation of Britain’s relationship with the rest of the EU last year and won some changes, including the right to gradually phase in benefits for new arrivals from other member states. But that deal lapsed when the public voted to leave and the rest of the EU is now keen to hear what Britain’s demands will be for the article 50 process, which could take up to two years and which all countries must be willing to sign up to.

May was given a military welcome in Paris for the talks, which lasted around an hour. After the press conference, May and Hollande attended a working dinner where the leaders were served lobster salad, veal with spinach, vanilla mousse with strawberries and cheese.