Witnesses said the wall of water triggered by the 8.0-magnitude quake was up to five metres (16.
5 feet) high and swamped towns, flattened houses and washed away buildings in the west of the archipelago.
"Some villages are completely wiped out," Fred Fakari, the chairman of the Solomons National Disaster Council, told journalists in the capital Honiara.
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"Unconfirmed reports have placed the number of dead at about six and I think we are expecting more," Prime Minister Manasseh Sogavare told CNN.
"It's a very trying time for us.
"
He said patchy communications made it difficult to assess the scale of the casualties and damage in Gizo, a popular tourist and diving spot that is just 40 kilometres (25 miles) from the epicentre.
"Most of the islands are low-lying, and the extent of the damage is yet to be known," he said, adding that a team from his office, the Red Cross and the police were headed to the area.
The Pacific Tsunami Warning Centre issued a tsunami warning for the regi