Peers have dealt a major blow to Theresa May’s Brexit strategy by backing a bid to force the government to effectively keep the UK in the single market.

Rebels in both Labour and Tory ranks defied the leadership to inflict the defeat on the government in the Lords, where they supported a cross-party amendment calling for continued participation in the European Economic Area (EEA).

The news will come as a blow to Jeremy Corbyn, as 83 Labour peers rebelled against a frontbench edict to abstain on the vote. Some 17 Tories also defied Ms May to back the bid.

The vote comes as the government suffered two other losses when peers backed plans to remove references to an official Brexit date from the face of the bill as well as plans to keep the UK in EU agencies after Brexit.

Later, there was a fourth defeat over calls to strengthen scrutiny of secondary legislation made by ministers following Brexit - taking the total number of government defeats to 14.

Labour’s Lord Alli, moving the amendment, said: “It is the EEA that deals with services, services like retail, tourism, transport, communications, financial services and aerospace where we have a £14bn trade surplus in these services.

“The customs union will benefit our European neighbours in their imports, and without an EEA equivalent it will damage our profitable export business and therefore the jobs and livelihoods of many thousands of people.

“It’s for that reason that we need to ensure that any continuation of the customs union must include a continuation in the EEA or its equivalent.”

Responding for the government, Brexit minister Lord Callanan warned that remaining in the EEA “would not deliver control of our borders or our laws”.

He said: “On borders it would mean that we would have to continue to accept all four freedoms of the single market, including freedom of movement.

“On laws it would mean the UK having to implement new EU legislation over which in future we will have little influence and, of course, we will have no vote.

“This will not deliver on the British people’s desire, as expressed in the referendum, to have more direct control over decisions that affect their daily lives.”

Labour rebels backing the amendment included former party leader Lord Kinnock, Lord Mandelson and ex-cabinet minister Lord Hain. The Tory rebels included former cabinet minister Lord Patten of Barnes and former deputy prime minister Lord Heseltine.

Earlier, peers backed a bid to remove the official exit date, 29 March 2019, from the bill, which is likely to infuriate Eurosceptics who see the bid as an attempt to thwart Brexit.

Labour peer Baroness Hayter said stripping the date from the bill would ”remove the straitjacket” and make the task easier for negotiators.

She told peers: “If this amendment is successful it will remove the straitjacket that the government are in – I have to say not at the behest of negotiators but at the behest of certain ardent Brexiteers.”

The efforts were spearheaded by the Conservative politician Charles Wellesley, Duke of Wellington, who insisted he was not trying to “thwart the process” of leaving the EU.

“We know beyond any doubt that for the purposes of this bill we leave the EU on 29 March 2019,” he said.

Brexit threatens life on the Irish border: in pictures Show all 15 1 /15 Brexit threatens life on the Irish border: in pictures Brexit threatens life on the Irish border: in pictures An abandoned shop is seen in Mullan, Co Monaghan. The building was home to four families who left during the Troubles. The town was largely abandoned after the hard border was put in place during the conflict. Mullan has seen some regeneration in recent years, but faces an uncertain future with Brexit on the horizon Reuters Brexit threatens life on the Irish border: in pictures A defaced ‘Welcome to Northern Ireland’ sign stands on the border in Middletown, Co Armagh Reuters Brexit threatens life on the Irish border: in pictures Mervyn Johnson owns a garage in the border town of Pettigo, which straddles the counties of Donegal and Fermanagh. ‘I’ve been here since 1956, it was a bit of a problem for a few years. My premises has been blown up about six or seven times, we just kept building and starting again,’ Johnson said laughing. ‘We just got used to it [the hard border] really but now that it’s gone, we wouldn't like it back again’ Reuters Brexit threatens life on the Irish border: in pictures Farmer Gordon Crockett’s Coshquin farm straddles both Derry/Londonderry in the North and Donegal in the Republic. ‘At the minute there is no real problem, you can cross the border as free as you want. We could cross it six or eight times a day,’ said Crockett. ‘If there was any sort of obstruction it would slow down our work every day’ Reuters Brexit threatens life on the Irish border: in pictures John Murphy flies the European flag outside his home near the border village of Forkhill, Co Armagh Reuters Brexit threatens life on the Irish border: in pictures Potter Brenda McGinn stands outside her Mullan, Co Monaghan, studio – the former Jas Boylan shoe factory which was the main employer in the area until it shut down due to the Troubles. ‘When I came back, this would have been somewhere you would have driven through and have been quite sad. It was a decrepit looking village,’ said McGinn, whose Busy Bee Ceramics is one of a handful of enterprises restoring life to the community. ‘Now this is a revitalised, old hidden village’ Reuters Brexit threatens life on the Irish border: in pictures Union Flag colours painted on kerbstones and bus-stops along the border village of Newbuildings, Co Derry/Londonderry Reuters Brexit threatens life on the Irish border: in pictures Grass reflected in Lattone Lough, which is split by the border between Cavan and Fermanagh, seen from near Ballinacor, Northern Ireland Reuters Brexit threatens life on the Irish border: in pictures Donegalman David McClintock sits in the Border Cafe in the village of Muff, which straddles Donegal and Derry/Londonderry Reuters Brexit threatens life on the Irish border: in pictures An old Irish phone box stands alongside a bus stop in the border town of Glaslough, Co Monaghan Reuters Brexit threatens life on the Irish border: in pictures Billboards are viewed from inside a disused customs hut in Carrickcarnon, Co Down, on the border with Co Louth in the Republic Reuters Brexit threatens life on the Irish border: in pictures Seamus McQuaid takes packages that locals on the Irish side of the border have delivered to his business, McQuaid Auto-Parts, to save money on postal fees, near the Co Fermanagh village of Newtownbutler. ‘I live in the south but the business is in the North,’ said McQaid. "I wholesale into the Republic of Ireland so if there’s duty, I’ll have to set up a company 200 yards up the road to sell to my customers. I’ll have to bring the same product in through Dublin instead of Belfast’ Reuters Brexit threatens life on the Irish border: in pictures A disused Great Northern Railway line and station that was for customs and excise on the border town of Glenfarne, Co Leitrim Reuters Brexit threatens life on the Irish border: in pictures Alice Mullen, from Monaghan in the Republic of Ireland, does her shopping at a former customs post on the border in Middletown, Co Armagh. ‘I’d be very worried if it was a hard border, I remember when people were divided. I would be very afraid of the threat to the peace process, it was a dreadful time to live through. Even to go to mass on a Sunday, you’d have to go through checkpoints. It is terribly stressful,’ said Mullen. ‘All those barricades and boundaries were pulled down. I see it as a huge big exercise of trust and I do believe everyone breathed a sigh of relief’ Reuters Brexit threatens life on the Irish border: in pictures A bus stop and red post box stand in the border town of Jonesborough, Co Armagh Reuters

“But this date should not be defined and specified ... in case it becomes necessary and in the national interest to agree an extension as provided in Article 50.”

Lord Callanan told peers that he saw “no reason” to amend the bill, adding: “I would reiterate that exit day within the bill does not effect our departure from the EU, which is a matter of international law under the Article 50 process.”