A spokesperson for Theresa May said: "The EU's provisions for visa-free travel into and out of the Schengen area cover Gibraltar, and mean that in any scenario, British Nationals from Gibraltar will be able to travel for short stays in and out of Spain and other countries in the Schengen area".
The seven square-kilometre peninsula is an internal, self-governing British overseas territory.
Hammond also noted that the United Kingdom had always had a vision of itself as a part of the world, not just Europe, adding that the country was well positioned to develop trade relations with various expanding markets.
The contingency work for no deal "is intensifying both within the European Commission and across the member states", according to the Irish Government.
"There is a controversy between Spain and the United Kingdom concerning the sovereignty over Gibraltar, a territory for which a solution has to be reached in light of the relevant resolutions and decisions of the General Assembly of the United Nations".
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In a 2002 referendum on whether Britain and Spain should share sovereignty over Gibraltar, 99 percent of voters said no.
Two-thirds of exporters to the European Union were looking to relocate overseas, and just under half of IoD members who are engaged with contingency planning have examined the feasibility of moving operations.
Separately, May's office rejected a report in the Mail on Sunday newspaper that her advisors were considering an early election on June 6 if she got a Brexit deal through parliament.
The row highlighted strains over the territory on Spain's southern coast as Britain quits the Union and the remainder of the EU states swing the bloc's policies behind Spain, which is taking advantage of Brexit to ramp up its claim to The Rock.
The Gibraltar Government said the worldwide legal status of Gibraltar was not a matter of argument, but rather of fact.
Spain had sought a more sweeping reference to United Nations resolutions on colonies, diplomats said, but France objected to protect its interests in some of its own overseas territories.
The BBC's Brussels reporter Adam Fleming said the Spanish government sees this as giving it a bigger say in the status of Gibraltar, and the issue is likely to be raised again as Brexit approaches, and beyond.