American credibility is in serious question here as the Lebanese sink deeper into despair just a week after they euphorically thought they were on the threshold of political stability for the first time in four years.

U.S. prestige is taking a battering because Washington is said to have promised the vulnerable Lebanese government that Israel would refrain from troublemaking if a 500-man Lebanese Army battalion was deployed in southern Lebanon, near the Israeli border.

The troops were sent there last week. Nevertheless, according to published reports here, which have taken on a life of their own despite denials in Washington and Tel Aviv, Israeli heavy artillery opened fire on them. In some cases, according to these reports, Israeli used American-provided anti-personnel ammunition proscribed for use outside Israel proper, and also fired on U.N. peacekeeping forces.

More damaging was the decision by Maj. Saad Haddad, the right-wing Christian militia commander in the south who is entirely dependent on Israel for arms, ammunition and food, to declare his six-mile-deep border strip an independent "free Lebanon."

Israel was credited here with having outsmarted and humiliated Lebanon and the United States. Is was a near return of last summer's disastrous efforts to deploy Lebanese troops in the same area, an earlier American-designed exercise sabotaged at the last minute by Israel, acting through Haddad.

Richard Parker, then U.S. ambassador to Lebanon, so lost the confidence of the Lebanese government - or at least that was the local reading - that he was reassigned to Morocco.

This time, many Lebanese are wondering about American ability to deliver the Israelis and suggest the United States may have connived in Haddad's declaration of a Free Lebanon.

Many Lebanese are convinced these recent events are interconnected:

Chances of a Moslem-Christian reconciliation to heal the wounds of the 1975-1976 Lebanses civil war evaporated when Former president Camille Chamoun, the Cleverest Christian politician, Saturday proclaimed his support for Haddad. That scuttled well-advanced plans for a coalition government to be formed once the Lebanese battalion was in the south.

The Syrians, infuriated by Chamoun's open espousal of Haddad and, by extension, of Iseael, reacted by shelling Ain Rummaneh, a Christian community, 24 hours later.

The 30,000-man Syrian truce force was also said by its Lebanese critics to have encouraged partisans of former president Suleiman Franjieh to kill more than a dozen Christian Phalangists Sunday.

The Phalange last June killed Franjieh's son, Tony; his daughter-in-law and granddaughter. Although the former president has sworm he will have Bechir Gemayel, the younger son of the Phalange leader, killed in vengeance, the Syrians previously kept Franjieh restrained behind a buffer zone of troops.

Israeli Prime Minister Menachem Begin said in parliament that Israel would defend the nothern Lebanese Christians as well a sHaddad's villagers in the souther border area.

Coupled with Haddad's proclamation of independence, Begin's remarks were seen by pessimists as consecrating the first step toward the formal partition of Lebanon that Chamoun and other Christian radicals have long threatened unless they got their way throughout the country.

Many analysts believed the Begin government was reacting to sustained American and other Western pressure to have the Lebanese Army present in the south.

That deployment was the condition laid down by governments providing troops for the 6,000-man U.N. peacekeeping force in the south for renewing their commitment to the year-old venture.

Israel has made no secret of its displeasure with the peacekeeping force, which effectively prevents the Jewish state from acting at will beyond Haddad's border strip.

Before Israel's invasion of southern Lebanon in March, 1978, Israeli troops faced no such international buffer and thus potentially threatened not only Palestinian commandos, but more importantly the soft underbelly of Syria itself.

Analysts believe the recent increased Syrian and Palestinian terrorist activity abroad reflect their desire to prevent the Egyptian-Israeli peace from gaining automatic acceptance internationally.

If for no other reason, Syrians and Palestinian commandos allowed the Lebanese Army battalion to bolster the U.N. force in the south.

Chamoun and other Christian radicals, however, seem determined to show their pro-Israeli hand, which they sought to minimize in the past.

They are fond of quoting the late Israeli prime minister David Ben Gurion, who long ago predicted that, although he did not know which Arab state would be the first to recognize Israel, Lebanon would be the second.

Now Egypt, the Arab world's largest and most influential state, has become the first to sign a peace treaty with Israel.

By the kind of dangerous comparisons popular in the Middle East, that would make Egypt, Israel and their American backers, allies of radical Labanese Christians.Official Washington has long insisted thes was not the case, however.

Expressing common Lebanese depression at the events of the past week, one well known editor said, "In Lebanon there are no longer any Lebanese - just Israelis and Syrians." CAPTION: Picture, A civilian car rushes for safety on the road between Beirut and Sidon in Southern Lebanon as Israeli-backed Christian militia shelled Palestinians. AP