The branch and two other banks had increased security since last week's crime, he said.
Mary Tini, 73, has lived on Aitutaki for most of her life, and said the crime had upset residents.
Many preferred to think an outsider, rather than a local, was responsible.
"It's the first time something like this has happened here. Police have been here talking to people and they had to close the bank for a few days.
"People are very shocked."
The island's mayor, John Baxter, told Radio New Zealand International he thought the burglary was not done by one of the 1800 locals, most of whom kept their savings at the bank.
"It's a very sad occasion or event that has happened.
"I think it is the first time any of the banks have ever been robbed on this small island ... Everybody knows everybody and I suspect whoever has done this does not live on the island."
As one of the first places in the South Pacific to accept Christianity, Aitutaki, about 250km north of the main island of Rarotonga, has always been celebrated for its piousness.
The 18sq km island has more than 20 churches representing more than a dozen denominations, including Protestant faiths, Mormons, Catholics, Seventh-day Adventists, Baha'is, Apostolics, the Assembly of God and Jehovah's Witnesses.
Islanders are so devout that tourists cannot arrive or leave the island on a Sunday and no tourism activities operate for the day.
Church leaders said the bank theft had come as a shock, and they were disappointed.
The president of the Seventh-day Adventist Church, Pastor Eliu Eliu, said Aitutaki - and the Cook Islands overall - was a very religious place.
Many people thought an outsider - a tourist or some other person not from Aitutaki - had burgled the bank.
But others suggested only a resident would know their way in the building.
"If it were a local, then we would be very concerned. What was his motive? It would probably have a lot of impact on the community, how they will feel about Christian values.
"It will also challenge our view of our teenagers today - asking whether this young person was simply expressing his frustrations about something happening in his life."