The shooting happened just after 1 a.m. at The Starboard restaurant in Dewey Beach.
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Authorities say employees called to report a man with a gun.
When they arrived, officers talked with the man, who has since been identified as Rodney K. Robinson II, 21, of Milford. Officials said Robinson then took off running.
About an hour later, a Starboard Restaurant employee called police again to report the suspect had returned.
During a foot chase, an officer fired at Robinson, hitting him, according to reports.
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He later died at the hospital.
So far, no further details have been provided.
Police are asking anyone who may have witnessed the incident or anyone with video surveillance to please contact Sgt. Stephen Yeich by calling 302-741-2703.