“The government, with its explicit policy of creating a hostile environment, is not having the intended effect of protecting the U.K., but it is having the opposite effect of undermining core institutions such as universities,” Professor Franklin said, citing a strategy put in place in 2012 by Theresa May, who was the home secretary at the time, that was meant to create “a hostile environment for illegal migration” in Britain.

Ms. Islam, 31, is originally from Aligarh, in northern India, and has lived in Britain since 2009. She has a string of academic successes to her name, including an award from the London School of Economics for her master’s degree and a Gates scholarship for her Ph.D. at Cambridge on gender and class in urban India.

Ms. Islam, whose student visa expires at the end of January, said she had been shocked and very upset by the rejection and would appeal the decision.

“For me, there are personal, professional, emotional and financial consequences that I am still in the process of taking account of,” Ms. Islam said. “Living with this kind of uncertainty is not nice at all.”

The Home Office did not comment directly on her case but said that each application was considered on an individual basis. The agency said that applications for permanent residency would be refused if an individual exceeded the 540-day limit.

Ms. Islam found out about the rejection last week and decided to speak out about her case by posting on Twitter.

Outrage about the decision spread quickly among the academic community and by Monday, the open letter had garnered more than 1,000 signatures. Ms. Islam said the outpouring of support was “heartening.”