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It is believed that Mr Amess has been receiving treatment at the scene for his injuries. His condition has not been confirmed.
Counter-terror police are involved in the early investigation into the stabbing and are working to establish the potential motivation.
There has not yet been any confirmation of whether the incident is being considered terror-related.
Police were called to the scene shortly after noon on Friday and said the surrounding roads have been closed off.
A spokesperson for Essex Police said: “We were called to reports of a stabbing in Eastwood Road North, Leigh-on-Sea shortly after 12.05pm today.
“Officers were on scene shortly after, arrested a man and recovered a knife. He is currently in custody.
“We are not looking for anyone else in connection with the incident and do not believe there is an ongoing threat to the wider public.”
Local Conservative councillor John Lamb told Reuters Sir David was stabbed several times, adding: “We're not sure how serious it is but it's not looking good.”
He said the MP had not been taken to hospital, but that there were two ambulances at the scene and ”they've been working on him all that time here.”
One witness called the incident “very distressing” and described seeing someone being taken out of the church and put into the back of a police car, before he was asked to leave the area.
“At that point obviously we knew something serious had happened,” he told LBC radio.
“Within five minutes we'd been informed by someone who was actually in the building what had happened and apparently he (Sir David ) had been stabbed quite a few times. It's very distressing.”
Sir David had advertised the surgery, which is a regular opportunity for constituents to air their views and seek assistance, on his Twitter page on Tuesday, giving the location and contact details to book an appointment.
Pictures taken of the scene of the incident showed a police car and multiple ambulances parked on the road, with the area cordoned off to the public.
Brendan Cox, the widower of murdered Labour MP Jo Cox, wrote on Twitter: “Attacking our elected representatives is an attack on democracy itself. There is no excuse, no justification. It is as cowardly as it gets.”
Politicians reacted to the news on Friday, with Labour leader Keir Starmer saying: “Horrific and deeply shocking news. Thinking of David, his family and his staff.”
Former prime minister David Cameron tweeted: “Very alarming and worrying news reports coming from Leigh-on-Sea. My thoughts and prayers are with Sir David Amess and his family.”
Conservative MP and former party leader Sir Iain Duncan Smith wrote: “My thoughts are with David Amess MP and his family at this awful time.
“Praying for a full recovery following this appalling, shocking news. This angry, violent behaviour cannot be tolerated in politics or any other walk of life.”
East Ham MP Stephen Timms, who was stabbed at a constituency surgery in 2010, wrote on Twitter: “Appalled to hear of the attack on amessd-southend today. I know him well and am thinking of him with very best wishes as we await further news.”
Sir David was first elected to parliament to represent Basildon in 1983, and then stood for election in Southend West in 1997.
He is the current chair of the All-Party Parliamentary British-Qatar Group, and recently met the country's emir in Doha.
On his website, he lists his main interests as “animal welfare and pro-life issues”.
Essex Police said anyone with information should contact the force quoting incident 445 of 15 October.
Additional reporting by PA