The man, aged in his 60s, was hit by a falling tree branch in Black Wood, Woolton, Liverpool, on Tuesday and later died in hospital.
A police spokesman said the dog, who was not injured in the accident, was still at the man’s side when emergency crews arrived at the scene.
The incident comes after a driver was killed when a tree fell on to his car as Storm Ciara battered the UK at the weekend.
The 58-year-old driver, from Micheldever, in Hampshire, was pronounced dead at the scene on the A33 after the tree toppled as he was driving his Mercedes on Sunday afternoon.
More stormy weather is in store, with the Met Office predicting strong winds and heavy rain from the next named weather system – Storm Dennis.
While the 97mph top recorded windspeed from Storm Ciara, brought to the Isle of Wight, is unlikely to be repeated, gusts of 50mph are expected, with more than 60mph possible over hills, coastal areas and exposed locations, the forecaster said.
The predicted heavy rain would also run the risk of further flooding.
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Around 730 properties are believed to have been affected by flooding in recent days, the Environment Agency said as it warned of more wet weather ahead when Storm Dennis takes hold.
Caroline Douglass, flood duty manager at the agency, said: "River and surface water flooding is possible on Saturday into Sunday due to Storm Dennis and we are advising people to check their flood risk and to stay safe on the coast or when walking or driving near swollen rivers.”
She also warned people against taking "dangerous" so-called storm selfies.
The Met Office said transport delays and cancellations can be expected as well as damage to power supplies and large coastal waves.
Chief meteorologist Steve Ramsdale said: "Another spell of very wet and windy weather is expected for Saturday.
"Although Storm Dennis is currently not expected to be as severe as Ciara, disruption is still likely.”
Bad weather continues
The clear-up operation continued as the rain and wind from Storm Ciara gave way to snow and ice across the country.
Heavy snow fell in some regions after the storm disrupted flights, ferries and trains throughout the UK and the Republic of Ireland.
On Tuesday there was an amber warning for heavy snow showers in southern Scotland as well as a yellow warning for wind and snow across much of the country.
That warning also applies to Northern Ireland, and a yellow warning for snow and ice is in place in parts of northern England.