The water temperature was a frigid 53 degrees. The young woman had been clinging to a tree limb in the surging Potomac River for more than half and hour. And now she could no longer feel her legs.

Logs and limbs swept past her on the frothing brown torrent, and as a television helicopter camera beamed the dramatic scene live Monday afternoon, the huge, flooded river seemed ready to swallow her.

Suddenly, at 2:04 p.m., a blue Montgomery County swift-water rescue boat appeared, battling upstream against the raging current. “Hurry up!” the woman cried. “I need to get out of the water!”

As the boat approached, firefighter Peter Gillis, wearing a helmet and an orange “dry suit,” leaned off the bow and swatted the tree branches aside. “Hang on,” he said. “We’re right behind you.”

Then he grabbed her. “I have you,” he said. “Let go.”

She seemed to hesitate, and he insisted: “Let go.”

Cold and exhausted, the 31-year-old kayaker from Woodbridge released her grip and was yanked into the rescue boat by Gillis and Master Firefighter Chad Pollard, ending a struggle that the boat crew said the river often wins.

Water from the weekend downpours swelled the Potomac on Monday, pushing it over its banks from Harpers Ferry to Georgetown and prompting a day of rescues and evacuations.

Twenty Boy Scouts and adult chaperones were rescued early Monday after they were trapped by rising water along the Potomac at White’s Ferry, Montgomery County fire officials said.

About 6:15 a.m., rescue teams used a helicopter and a boat to pluck the Scouts from a campground that had become cut off from the C&O Canal by rising water, fire officials said. No one was injured.

The Scouts had stopped there for the night during a hiking trip, fire officials said.

Just before 4 a.m., four campers were rescued from waist-deep water at Marble Quarry Campground in Dickerson, fire officials said. The group’s campsite on the river began to flood, and they tried to move to dry land but ended up in deeper water, fire officials said. The group was rescued by boat.

Floodwaters also poured into the riverfront Washington Harbour complex in Georgetown, swamping shops, restaurants and a parking garage and forcing the evacuation of businesses.

A murky brown tide 10 to 12 feet deep spilled into ground-level businesses and basements and submerged cars in the parking garage.

Pete Piringer, spokesman for the D.C. fire and rescue department, said the fire department was “quite surprised” that the flood walls were not up when crews arrived.

“I think the wall being up would have prevented this from happening,” Piringer said.

The management company of Washington Harbour, identified by news organizations as MRP Real Estate Services, was reported by news outlets as saying in a statement: “We have taken precautions to protect the residents, tenants and visitors to Washington Harbour . . . and are working expeditiously to mitigate further damage.”

Katherine Siahaan, a worker at an ice cream shop in the mixed-use development off K Street, said she and her colleagues didn’t expect flooding to be a problem at first, but the river “kept rising.” By 10 a.m., when Gelateria Dolce Vita was evacuated, barstools and a cash register were floating around the shop, Siahaan said.

“It was deep enough to drown a man,” she said.

But what appeared to be the most dangerous incident happened upstream near Violet’s Lock in Montgomery County.

Assistant Chief Scott Graham, a spokesman for the Montgomery County Fire and Rescue Service, said the rescued woman, whom he declined to identify, was taken to a local hospital suffering from hypothermia, dehydration and fatigue.

He said she had gone kayaking on the river just upstream at Riley’s Lock at Seneca Creek State Park. The kayak capsized, and she was carried downstream until she grabbed the tree branch. She had lost her helmet but was wearing a life vest.

Graham said rescue officials got a call about 1:15 p.m. from a bird watcher on the C&O Canal towpath who reported seeing a person clinging to a tree in the river about 60 yards off the Maryland shore.

Rescue crews hurried to the towpath, but the woman was too far out for the crew’s rope-filled “throw bags” to reach her. The rescuers were able to call to her and tell her to hold on with her legs, rather than just her arms.

Meanwhile, the swift-water rescue team was summoned.

The firefighters put the inflatable boat in the water at Violet’s Lock. With Capt. Joe Bell at the helm, they headed downstream, passed the victim, and made a U-turn to perform the rescue against the current.

As Bell eased the boat closer, careful to avoid debris, Gillis and Pollard hauled the victim aboard.

It could easily have turned out badly, Bell said. “Usually when we get to them, they’re long gone.”

Back on land, Pollard walked the woman to an ambulance.

“The day,” he told her, “can only get better.”

Staff writer Patricia Sullivan contributed to this report.