British forces shot down seven more Argentine warplanes in furious battles over the waters of the Falkland Islands yesterday, according to Britain's Defense Ministry, as Argentina's defense minister warned that the war could spread beyond the two countries.

An Argentine military communique issued late last night confirmed for the first time that British forces had succeeded in establishing a beachhead near San Carlos, on East Falkland Island, measuring 54 square miles and defended by about 2,000 men.

Waves of Argentine Mirage and Skyhawk aircraft attacked British vessels and shore positions on the Falklands, British accounts said, adding that at least seven were brought down by missiles and antiaircraft fire from ships, Harrier warplanes and shore batteries deployed since the first British landings on the islands Friday.

Britain announced the loss of the frigate Antelope, which sank after it was gutted by fire from Argentine bombing raids Sunday, with one British sailor reported killed and seven wounded. It was the third British warship lost in the seven-week-old war. A British military spokesman said yesterday's fighting may have caused additional damage to vessels of the task force.

Last night's Argentine military communique said that a British troop ship and a frigate were damaged yesterday by Argentine warplanes attacking British ships and shore positions. The communique said one Sea Harrier jet was downed during a British bombing raid near Port Stanley. It said two Argentine planes had been shot down. The Argentine Embassy in Washington said one of the damaged British ships was the troop ship Canberra.

Argentina's military leadership appeared to be staying with its strategy of wearing out British forces with its air raids and, amid reports of continued heavy losses, it indicated it would quickly look abroad for military support if its own forces grew thin.

Argentine military officials said that they were successfully encircling British forces and denying them crucial supplies.

The reported heavy fighting yesterday--and unconfirmed reports that Britain has captured an important Argentine airstrip at Goose Green on East Falkland--came amid diplomatic developments and hard-line declarations by both sides.

In a message to Pope John Paul II, who is scheduled to visit Britain Friday, British Prime Minister Margaret Thatcher rejected an appeal for a cease-fire. She said that unless the Argentine forces withdrew first, a truce "would leave the aggressor . . . in possession of the rewards of military adventure."

The pope had called for a cease-fire yesterday in letters to Thatcher and Argentine President Leopoldo Galtieri. Wire services quoted Vatican officials as saying that the scope of the six-day visit will be reduced because of the Falklands fighting, but the trip will not be called off.

British Defense Secretary John Nott told Parliament yesterday that "there can be no truce," and he said that the "days of the occupying Argentine garrison are numbered." He warned, however, that Britain "may still have a very difficult fight on its hands."

Argentine Defense Minister Amadeo Frugoli called Britain's counterinvasion of the Falklands, which Argentina calls the Malvinas, "a suicidal and colonialist adventure." He told a Buenos Aires radio station that the conflict "runs the risk not only of becoming more serious, but also of internationalization."

Frugoli said nothing about reports in the Buenos Aires press that Argentine military officials are becoming more willing to seek help from the Soviet Union. But he said that the United States, as the most powerful Western nation, "must think seriously about the gravity of the situation and adopt a more balanced position."

In Washington, President Reagan said that so far the Falklands fighting presented no reason for him to cancel a planned visit to London during a 10-day European trip next month.

At a meeting in Belgium, Britain and seven other European Community countries--but not Italy and Ireland--extended indefinitely a ban against imports from Argentina. The ban, imposed April 17 for a month, had been extended last week but only for seven days.

The U.N. Security Council met for the fourth day to debate the Falklands crisis as four countries--Panama, Ireland, Uganda and Japan--circulated draft resolutions urging a resumption of mediation efforts by Secretary General Javier Perez de Cuellar. But so far no formula has emerged that could satisfy both Britain and Argentina.

Argentina asked the Organization of American States to reconvene a conference of foreign ministers on the Falklands crisis to consider "further measures" against Britain.

The continued sea and air battles reported by Britain yesterday came as the skies suddenly cleared in the South Atlantic. The U.S. Weather Service said a storm had passed the area, leaving the skies "completely clear."

By Britain's count, at least 30 Argentine planes have been shot down in the past four days and 63--more than one-fourth of Argentina's entire pre-invasion Air Force--have been downed or destroyed on the ground since the British task force arrived in the South Atlantic late last month.

The Antelope, a 3,250-ton frigate, burned and sank after a 500-pound bomb exploded as British forces were trying to defuse it. The bomb had smashed into the ship's engine room during an Argentine raid Sunday. The crew had been taken from the vessel before it sank.

It was the third ship of the task force lost to combat, but Nott told Parliament that "more, highly capable" warships have reinforced the fleet.

Nott also announced the loss Sunday night of a Harrier "in an accident" during takeoff from a carrier. It was the fifth Harrier acknowledged lost by the British.

British newspapers reported the capture of a heavily defended Argentine airstrip at Goose Green, 20 miles south of San Carlos, where Britain has established a beachhead.