More than 1,000 members of the British military have deserted the armed forces since the start of the 2003 Iraq war, the BBC has discovered.
It comes as Parliament debates a law that will forbid military personnel from refusing to participate in the occupation of a foreign country.
Some 900 have evaded capture since the Iraq war started, official figures say.
But the Ministry of Defence says the numbers going missing from the army have stayed constant in recent years.
MoD figures show 2,670 soldiers went “absent without leave” in 2001, with the figure rising to 2,970 in 2002 and falling in 2003 to 2,825. In 2004 it rose to 3,050, falling back again in 2005 to 2,725.
But BBC world affairs correspondent Jonathan Charles said going absent without leave and desertion were not the same, adding a soldier who has gone absent without leave for more than 30 days could be considered as deserting.
He says figures showed 377 people are still missing after deserting during 2005 alone, with another 189 on the run so far this year.
And Labour MP John McDonnell said “I think what the MoD is saying flies in the face of all the other evidence and the experience of soldiers on the ground.”
He believes there are “a lot more seeking to avoid service, through different mechanisms”.
On Monday Mr McDonnell told Parliament the number of desertion cases had tripled over the past three years.
The MoD said it defined desertion as “going absent intending not to come back”, or “going absent to avoid active service”.
A spokeswoman said the soldiers currently missing were considered to be “absent without leave”, not to have deserted. They would have to be court martialled before they could be found guilty of deserting the army.
She said only one person had been found guilt of deserting since 1989.
Former defence minister Don Touhig told BBC Radio Five Live there were no “hard facts” to suggest the Iraq conflict was prompting increased numbers to leave the forces.
Our correspondent says it is unclear how many troops are deserting because they do not want to go to Iraq and how many are doing so for personal reasons such as family problems.
Lawyers say they are increasingly being contacted by people wanting advice about getting out of having to serve in Iraq, even if they do not want to desert, our correspondent adds.
Justin Hugheston-Roberts was the solicitor for Flight Lieutenant Malcolm Kendall-Smith who was sentenced to eight months in prison for refusing to follow orders in connection with a deployment to Iraq.
He said: “I am approached regularly by people who are seeking to absent themselves from service. There has been an increase, a definite upturn.”
Military law expert Gilbert Blades, who represents soldiers at courts martial, said the numbers leaving because of Iraq were often obscured as they were not counted as conscientious objectors.
“One can’t help thinking that what’s behind every absence is the problem in Iraq and I would think that if the real truth was told, then the Iraq problem has contributed to a huge number of people going absent,” he told BBC Radio Five Live.
Our correspondent says there is plenty of anecdotal evidence from military personnel that they are demoralised by the Iraq conflict and by the fact that, despite their best efforts, they see little improvement in the situation there.
Former SAS member Ben Griffin was allowed to leave the military after telling his commanding officer he was not prepared to return to Iraq because of what he believed were illegal acts being carried out by US forces.
He says: “I was disturbed by the general day-to-day attitude of the American troops. They treated Iraqis with contempt, not like human beings. They had a complete disregard for Iraqi lives and property.”
Mr Griffin would never have considered deserting but says his views are shared by many others in the British military.
He told the BBC: “There’s a lot of dissent in the Army about the legality of war and concerns that they’re spending too much time there”.
He says Iraq is different to other conflicts because, in other operations, the main aim is to improve life for the local population and he believes that is not what has happened in Iraq.
Mr Griffin says: “There’s contempt for the locals. We don’t even know how many have been killed.”
His advice to others is not to desert – but to follow their conscience and speak out if they think the conflict is wrong.
Major General Patrick Cordingley, who commanded the 7th Armoured Brigade Desert Rats in the first Gulf war, said servicemen’s views on Iraq prompted some to leave but “good leadership” would avoid it reaching epidemic proportions.
He said those who had been to Iraq before or whose families were unhappy about them going were among those who might not want to serve there.
“If you have such a person in your unit you have to discuss things with them… you do not necessarily want people with you if they have that particular view,” he added.