The embassy announced late on Sunday it was indefinitely shuttering its consular office in Kabul for security reasons after it said several staff were harassed by unknown assailants on roads in the Afghan capital.
The visa section's closure comes as there were fresh tensions between Afghanistan and Pakistan, which last week exchanged fire in a border area.
It also highlights the reliance many Afghans have on Pakistan, where they seek the type of medical care that is unavailable or unaffordable in Afghanistan.
"I am in a hurry to reach Pakistan, as the mother of my seven children is very ill, and I urgently need to take her to Pakistan for treatment," Abdullah Abded, who had travelled from Parwan province north of Kabul said.
"I feel desperate. I may have to tell my children that their mother may die because we cannot get the visa."
While the war-torn country has many health facilities, these are concentrated in the cities and access is often determined by wealth.
Specialist treatment in Kabul, for instance, can cost double what it would in Pakistan, people outside the embassy said.
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"I have a kidney disease and need to go to Pakistan on a regular basis for treatment or my kidneys will fail," said Nabi Yosufzai, a 26-year-old shopkeeper.
The consulate has closed in the past, usually for short periods. A sign on the consular section's door in Pashto on Monday said it had been closed until further notice.
The Foreign Office (FO) on Sunday summoned the Afghan charge d'affaires to "convey serious concerns over the safety and security of the diplomatic personnel of the embassy in Kabul, and its sub-missions."
In a statement late on Monday, the Afghan foreign ministry said it would "seriously investigate" the claim, but also expressed its "deepest objection" to the summoning of the Afghan envoy in Islamabad.
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An embassy spokesperson said that the the consular section gets about 2,000 applications daily, and is usually able to process about 1,500 visas within a 24- or 48-hour time frame.
Apart from for medical help, Afghans seek Pakistan visas to visit loved-ones, buy and sell goods and get university educations.
"I feed my big family by selling Afghan almonds or pomegranates in Pakistan in small amounts," said Kabul trader Nematullah, who only goes by one name.
"The profit is hardly enough for my family, but still better than the money you can make here."
Pakistani consular services remained open in Herat, Jalalabad and Mazar-e-Sharif.
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