House prices reached a record high of £208,000 on average in December after leaping by 9.5% during 2015, according to Halifax.
Across the UK, the average price in December was £208,286, after increasing by 1.7% month-on-month, it said.
The annual increase of 9.5% recorded in December was up from 9% in November.
But there are signs that the pace of price growth is softening slightly.
The quarterly rate of growth, which tends to be a good indicator of underlying trends, has remained below 2% for two months in a row, Halifax said.
The quarterly rate showed that prices in the three months to December were 1.6% higher than in the previous three months. In November, the quarterly rate was 1.4%.
Martin Ellis, a housing economist at Halifax, said the latest figures indicate a possible slight softening in the underlying pace of house price growth.
He continued: "There remains, however, a substantial gap between demand and supply with the latest figures showing a further decline in the number of properties available for sale.
"This situation is unlikely to change significantly in the short term, resulting in continuing upward pressure on prices."
Halifax pointed to recent figures from the Royal Institution of Chartered Surveyors (Rics), showing that the supply of homes on the market is at a record low.
Meanwhile, HM Revenue and Customs (HMRC) figures have shown that UK home sales during September to November were 5.2% higher than in the same three months a year earlier.
A "bumper backlog" of nearly half a million homes across England are still waiting to be built after being given planning permission, according to separate analysis released by the Local Government Association (LGA).
A study commissioned by the LGA found that 475,647 homes were yet to be built across England in the financial year 2014/15 after being given planning permission.
But the Government said that work has already started on more than half of these homes.