But immigration minister Kevin Foster has insisted the UK Government will “not be extending the deadline”, arguing EU nationals have had time to apply because the settled status scheme has been open since 2019.
Ms Sturgeon met with EU citizens in Edinburgh on Wednesday, alongside organisations supporting people applying for settled status.
Ms Sturgeon claimed the application process is forcing people who have made Scotland their home “jump through hoops”, and she warned it could lead to a situation similar to the Windrush scandal.
“The UK Government’s refusal to listen to our call to extend the deadline is unacceptable and means all EU citizens must urgently apply for settled status if they have not already done so,” Ms Sturgeon said.
“To all EU citizens my message is this – Scotland is a better country because you are here. You are part of us and we badly want you to stay.”
She said she was “utterly heartbroken” at the situation after speaking to Virginia Zamojski – known as Ginny Tate – a 31-year-old healthcare support worker who has spent the vast majority of her life living in Fife.
Ms Sturgeon told the PA news agency: “She came here from Germany at three years old, she’s been in the care system and she’s as Scottish as I am but she’s having to apply for the right to stay here.
“It’s absolutely despicable.
“These are people who work in our health service, our social care system and our food and drink sector.
“We need people to want to live and work in Scotland, and anything that puts a barrier in the way of that is not just wrong in principle but it’s actually undermining our own self interest.
“We will continue to press the UK Government to extend that deadline.
“It is unthinkable that they will have an absolute guillotine and then leave people who, for whatever reason, might not have applied or might not have provided all the right information to effectively lose their legal right to stay here.
“But this is the UK Government obsessed with Brexit that we know, so that can’t be ruled out.”
Ms Tate told PA she does not remember being in Germany and has never been back, but is living “in fear of being deported”.
She plans to send off the 40-page settled status application later this week that required bank and mortgage statements, and said: “Then it’s just a waiting game and I’ve got to see whether I get accepted or not.”
Ms Sturgeon compared the situation to that faced by the Windrush generation and said: “I don’t raise that lightly because that was a horrific way in which people were treated, who lived here for years and then found they didn’t have a right to be here.
“But there is a real danger we see a repetition of that if there isn’t great care taken and if this deadline is not extended.”
Asked whether the scenes in Glasgow’s Kenmure Street, where hundreds of people gathered to prevent immigration officers taking two men away, could become a more regular occurrence, Ms Sturgeon said: “The ball is in the court of the Home Office.
“If the Home Office continues to treat people with a lack of basic humanity and dignity, to try to move people out of communities that consider them part of the community, then it will not be a surprise if Scottish communities come together to protect people who live amongst them.
“I’m not complacent about racism and discrimination in Scotland, we are not immune from any of that, but I think there is a genuine feeling that people who have come to this country, who make a contribution here, who work here, who want this country to be their home, should be welcomed and encouraged to stay.
“That’s just out of altruism, although that shouldn’t be underplayed, but because we have a country that has a bit of a challenge for our own demographics over the next 20 years.
“So we really need to encourage people to come here.”