Can you be immune to a coronavirus?
It is hypothesized that exposure to and recovery from SARS-CoV-2 (as with other coronaviruses in humans) would generally result in short-term immunity to this strain, but we do not yet have data on this:

However, according to Dr Stephen Gluckman, an infectious diseases physician at Penn Medicine and the medical director of Penn Global Medicine, who spoke to the outlet, it seems likely that having the disease once results in immunity in most individuals - as is seen with other coronaviruses.
“Coronaviruses aren’t new, they’ve been around for a long, long time and many species - not just humans - get them,” he explained. “So we know a fair amount about coronaviruses in general. For the most part, the feeling is once you’ve had a specific coronavirus, you are immune. We don’t have enough data to say that with this coronavirus, but it is likely.”
This means that people who initially recovered are more likely to relapse rather than get reinfected with the virus.
According to one study, people with mild infections can test positive for the virus by throat swabs “for days and even weeks after their illness”.
But, that doesn’t mean it isn’t possible to contract the disease again, especially in those who are immunocompromised.
“The immune response to Covid-19 is not yet understood,” the Centers for Disease Control and Prevention (CDC) explains. “Patients with MERS-CoV infection are unlikely to be reinfected shortly after they recover, but it is not yet known whether similar immune protection will be observed for patients with Covid-19.”


Coronavirus: Can You Get Covid-19 Twice Or Does It Cause Immunity?

It is also hypothesized (in humans) that previous exposure to coronaviruses may enable immunity to certain other highly related strains:

These data indicate that challenge immunity to coronaviruses is strong, but highly virus strain-specific.


Virus strain specificity of challenge immunity to coronavirus., Archives of Virology (1989)

But these are all examples of adaptive immunity - not innate.