Police have issued a plea to dog owners to control their pets amid a series of alarming incidents in Richmond Park at the weekend.

The park is a destination for thousands of people from across the capital on spring weekends and many are bringing their dogs.

The crowds are becoming so large that the park has become a battleground between visitors, with long traffic jams to get into the car parks, cyclists, who race around the perimeter roads, dog owners and picnickers.

On Sunday alone, there were at least three incidents of dogs off their lead and out of control chasing terrifying young deer.

At the same time, there were reports on social media of dogs harassing deer, swans and other wildfowl in the ponds at Bushy Park.

The Royal Parks Police tweeted about the problems, saying: "We have had at least 3 incidents of dogs chasing deer in Richmond Park this morning.

"The park is very busy, the deer are being chased across roads which have many cyclists on. If they take out a cyclist or pedestrian, it could result in serious injuries.

"If you think your dog may chase the deer, keep it on a lead."

A video of one of the incidents has been viewed more than 250,000 times while many people responded to the tweet by criticising users of the park, including the dog owners, cyclists and motorists.

One visitor complained on Twitter, saying: "Richmond Park is a mad place , too many cars, cyclists, there are dogs, deer, horses, joggers, families, all in one space, all criss-crossing each other. It's nerve wracking, so I don't go there anymore."

The police video showed a small light-coloured dog racing after young deer, while the woman owner can be heard desperately shouting for the animal to stop.

The footage triggered a furious backlash with some people calling for court action and stiff penalties against the owners and some even suggesting the dogs should be shot, much as might happen with dogs worrying sheep on a farm.

One person said: "If a dog owner can't control the animal … they should be held accountable for its actions and face the law. At this time, stress might cause does and hinds to miscarry."

Another said: "We all need to look out for each other more and look after everyone safety. It won't be great for a deer to be injured by a car driver either as well as cyclists, children and other pedestrians put at risk by these dog owners."

Several people argued that it is time for a new byelaw requiring owners to keep their dogs on a lead at all times in both Richmond and Bushy parks.

One owner said: "It's such a shame the untrained dogs who can't be controlled ruin it for people who have trained their dogs well. My lab comes to an abrupt halt if we are off lead and come across deer and awaits instruction or for his lead to be put on

"If you can't keep your dog under close control and recall away from deer … then the dog should not be off lead."

One regular visitor said: "Need to change to rules so that all dogs are on leads. Can't picnic with the kids anymore there as off lead dogs always crash the party looking for food."

Another said: "Typical attitude of self-entitled pet owners. In National Park areas, dogs get shot by farmers for this kind of behaviour around livestock."

One more said: "The dogs in question should be confiscated and re homed with responsible people. It's not the dog's fault . The irresponsible owners should face stricter punishment/prosecution."