WELLINGTON, New Zealand—As New Zealand house prices surged, Niyaaz Sayed’s prospects of owning a home in the country he migrated to nearly two decades ago have become vanishingly small.

Mr. Sayed, a devout Muslim, says he would not buy a home using a bank mortgage because paying interest goes against an important tenet of his faith. That leaves him and his family with almost no option but to permanently rent in New Zealand, where the median house price shot up by about $7,500 a month in 2020 and 2021, far outpacing wage growth.

New Zealand is among a handful of smaller countries, including Sweden, Denmark, Norway and Finland, that have welcomed Muslim migrants in recent decades but haven’t developed a formal Islamic finance industry to support them. Islamic banks aren’t allowed to earn or pay interest and seek to offer products that comply with Islamic moral codes, known as Shariah.

Torkel Brekke, a professor of religion at Oslo Metropolitan University, said in a 2020 research paper that Nordic governments were unwilling to change regulations to accommodate Islamic finance.

In New Zealand, the finance ministry said it had “not produced any work in this area.” The country’s central bank said there are no regulatory hindrances to an Islamic bank applying for registration and it would welcome an approach for Shariah-compliant credit options to be explored. In the absence of a formal industry, at least one grass-roots group has stepped in to offer some interest-free loans, but the pace of house price growth threatens to stifle that effort.

A big constraint is the small size of Muslim communities in these countries relative to those in the U.S. and Australia, where Islamic finance has flourished. In New Zealand, there are about 60,000 Muslims in a country of five million.

Median house prices in New Zealand rose sharply in 2020 and 2021; a home for sale in Wellington.

Mr. Sayed, who estimated 80% of his Muslim friends in New Zealand would follow Islam’s prescription against paying interest, considered buying land that could take a modular “tiny home” and could be expanded over time. But soaring land values in Wellington, where Mr. Sayed works as an adviser to the human-resources chief at New Zealand’s tax department, scuttled that plan.

“We’ve become resigned to the fact that we are not going to own a home in New Zealand,” he said.

The impact of red-hot home prices is being felt across the community. Some Muslims have left New Zealand because they cannot get on the property ladder. Others, such as agricultural scientist Mustafa Farouk,

have ruled out moving to remote parts of the country where home values are lower because it would maroon them far from a mosque.

Mr. Farouk, who has worked in New Zealand for about two decades, worries that people who migrated to the country in recent years could face difficult futures. Property can act as a nest egg in retirement as people draw down equity to supplement their income.

Before the pandemic, New Zealand’s home prices were already among the most expensive for wealthy nations relative to incomes, partly due to constraints on development pushing up land prices. But emergency measures to shore up the economy, led by the central bank cutting interest rates to a record-low 0.25%, turbocharged the property market further.

Some second- or third-generation Muslims own homes they inherited and have benefited from rising values. But devout migrants who came to New Zealand more recently can face the prospect of saving the entire purchase price. That might have been possible 20 years ago, but now appears unfeasible even for a couple who both have high-paying jobs.

The difficulties of home ownership for New Zealand’s Muslims reflect the country’s broader failure to keep housing within reach of the average earner, said

Zainab Radhi, who runs a migrant-support organization.

Different approaches to housing finance, such as rent-to-buy Islamic models, should be considered to address the shortage of affordable homes, she said.

A possible way to amass the capital needed is for New Zealand’s government to sponsor the issuance of an Islamic bond, Dr. Radhi said.

Mr. Sayed, at the Wellington Islamic Centre and Masjid last week, said, “We’ve become resigned to the fact that we are not going to own a home in New Zealand.”

Tariq Ashraf, who investigated Islamic finance possibilities when he was a government official more than a decade ago, said 60,000 people might sound like a viable market, but isn’t when broken down by age, income and other factors that financial institutions consider.

There is no data on what proportion of Muslims in New Zealand wouldn’t use a mortgage on religious grounds. Some overseas research has indicated a link between higher education and the desire for Islamic financial services.

Software engineer Usama Salama hadn’t anticipated that securing Islamic finance to buy a home would be impossible when he migrated with his family to New Zealand from Egypt in 2014.

Local banks weren’t interested in working with the family on a lease-to-buy arrangement, Mr. Salama said. The family also considered contributing to a community-based program that makes interest-free loans, but it would have been years before they became eligible, he said.

As a result, Mr. Salama in 2019 moved to Melbourne, Australia, where four institutions, including one of the largest banks, had Islamic options for home purchases. Mr. Salama said he knew of three other Muslim families that left New Zealand because of the difficulty in buying a home.

In New Zealand’s largest city, Auckland, several dozen Muslim families contribute financially to a community-based program known as Sharif Financial Services that has provided interest-free loans for nine homes since 2012.

Fawad Sharif, one of the program’s founders, says its outstanding loans are 3 million New Zealand dollars, equivalent to $2 million. Members make regular financial contributions and a general meeting decides which loans to make. No interest is paid, but borrowers need a minimum 35% deposit and to pay the equivalent of around $10,000 to cover legal and administrative expenses.

Dr. Zainab Radhi, who runs a migrant-support organization in New Zealand, says other alternatives in housing finance, such as rent-to-buy Islamic models, should be considered.

Still, record house prices and factors such as the challenging financial situation for many people mean it could be another two years before Sharif Financial Services lends again, he said.

Faruk Balli, a professor of international finance at Massey University in Auckland, said even if Islamic finance for housing was available, many migrants wouldn’t be able to afford it.

Rising house prices also push up rents, which works against some rent-to-buy partnerships used in Islamic finance, he said. Repayments are based on a rental valuation of the property, which for the buyer functions best in a stable housing market.

“It’s not an easy country for an immigrant if you aren’t very high-skilled,” he said.

Write to Stephen Wright at stephen.wright@wsj.com