Nearly 20,000 men have fled Ukraine since the outbreak of war to avoid being drafted.
The BBC reports that Another 21,113 men attempted to flee but were caught by the Ukrainian authorities.
After Russia's invasion, most men aged 18-60 were banned from leaving. But data obtained by the BBC reveals dozens have made it out daily.
The BBC has established - by requesting data of illegal border crossings from neighbouring Romania, Moldova, Poland, Hungary and Slovakia - that 19,740 men illegally crossed into these countries between February 2022 and 31 August 2023.
The majority - 14,313 - were attempting to walk or swim across the border, and the remaining 6,800 relied on fraudulently obtained official paperwork stating fake exemptions such as fabricated illnesses, the Ukrainian authorities said.
The president's parliamentary representative, Fedir Venislavskyi, acknowledged to the BBC that the problem was serious.
"The government realises that this phenomenon is not isolated and that it is widespread. But unfortunately, I would emphasise that corruption is very resilient," he said - adding that Ukraine was doing "everything possible to keep the number of corruption cases to a minimum".
Mr Venislavskyi said the number of men who had left or had tried to leave was having no impact on the war effort.
"I am convinced that the resilience and readiness of Ukrainians to defend their independence, sovereignty and freedom is 95-99%," Mr Venislavskyi told the BBC.
"Those who try to avoid mobilisation are about 1-5%. They are definitely not critical to the defence of Ukraine." He said there were no plans to radically increase the number of those eligible for mobilisation.
The 40,000-plus number of men who have fled, or tried to flee, could represent a significant proportion of the men Ukraine needs to replenish its army. In August, US officials estimated the Ukrainian military death toll to be up to 70,000 - although Kyiv won't give a figure.
Read more at the BBC
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