European Commission President Jean-Claude Juncker has said the process should begin "immediately".
And several EU foreign ministers have urged Britain to start the process soon.
But Mr Hammond insisted that "nothing is going to happen at the moment".
The timing of the formal notification of intention to leave the EU was in Britain's hands and talks would not start until a new prime minister was chosen to replace David Cameron, he said.
Mr Hammond's position was supported by German Chancellor Angela Merkel's chief of staff, who said Britain's politicians should take time to review the consequences of leaving the EU.
"Politicians in London should take the time to reconsider the consequences of the Brexit decision - but by that I emphatically do not mean Brexit itself," Peter Altmaier told Reuters news agency.
Mrs Merkel said on Saturday that the EU had "no need to be particularly nasty in any way" in the negotiations with Britain, and that deterring other countries from leaving the bloc should not be a priority in the talks.
The EU has clarified the way the UK can kick start formal negotiations to exit the bloc following Thursday's referendum.
It says Britain can trigger Article 50 of the Treaty of Lisbon, which sets a two-year deadline for a deal, by making a formal declaration either in a letter or a speech at a meeting of the European Council, the gathering of ministers from each EU state.
UK PM David Cameron has said he will step down by October to allow his successor to conduct the talks.
Since Thursday's vote there has been intense speculation about when, and how, the UK might begin formal negotiations.
The UK's decision to leave the EU has sent shockwaves across the continent with leaders of Eurosceptic parties in France, the Netherlands and Italy demanding referendums in their own countries.
In response, some EU politicians have called for speedy reforms to quell further unrest.
French Economy Minister Emmanuel Macron suggested a new mission statement should be drafted and put to a referendum of all EU citizens.
"We've never had the courage to organise a true European referendum in its real sense," he told a conference.
"We would first build this new project with European peoples and then submit this new roadmap, this new project, to a referendum [across the bloc]."
The first summit of EU leaders with no British representation will be held on Wednesday, a day after Mr Cameron holds talks with members.
Global stock markets and the pound fell heavily on the news of the so-called "Brexit", while credit rating agency Moody's cut the UK's credit rating to "negative".
Brexit: What happens now?
Britain will resist pressure from the European Union for a swift start to negotiations on its withdrawal from the bloc, Foreign Secretary Philip Hammond has indicated.