Not if a substantial number of them now have antibodies against it.
More realistically, even if enough of them have antibodies to create herd immunity. (Via making it impractical for the virus to transmit to a vulnerable host)
The fraction of people needed to create herd immunity in a population is 1 - 1 / R0. With R0 at 4-7, that means about 75-86% of people within the population need to have COVID-19 antibodies before transmission naturally dies out through herd immunity. With current case numbers, China doesn't have anywhere close to this many cases - it would imply ~1B infections within the country rather than the ~80K that have been found so far, off by a factor of 10,000.
I think a likely outcome is that the current outbreak dies down, then it either flares up again from some undetected cases or is reintroduced from another country. But then, with the initial media attention elsewhere and COVID-19 symptoms basically looking like the flu, they'll just let the sick die rather than re-introduce draconian quarantine measures.
No, there is no solid evidence of persistent infections. The few reports of repeat infections are highly likely to be caused by testing artifacts. The imperfect sensitivity and false-positive rate of the tests means you'll always see people who tested negative at one time, and positive at a later time.
As an RNA virus, it would be extremely difficult for SARS-COV-2 to establish persistence. I'm not aware of any known mechanism by which that could happen. It's been proposed that some COV's may have neurotropic capability [1] that might allow them to achieve latency, but it's just that -- a proposal.
More realistically, even if enough of them have antibodies to create herd immunity. (Via making it impractical for the virus to transmit to a vulnerable host)