There's an investigation after reports a woman made threats to staff at a Tesco Express in Seaside Road.
It happened at around 7pm on Saturday, November 13. Security and staff asked the woman to leave the store but she continued to make threats against them.
She eventually left, walking off via Bourne Street.
Shortly afterwards officers arrested a 34-year-old woman from Eastbourne on suspicion of harassment, common assault, and racially aggravated common assault.
She was released on police bail until December 5 while the investigation continues.
PC Julia Turner said she wanted to hear from anyone who saw what happened or anyone who recorded mobile phone footage of the incident to come forward.
Anyone with information about the incident is asked to report it to Sussex Police online or call 101 and quote serial 1124 of 13/11.