The outbreak of the novel Coronavirus Disease 2019 (COVID-19) has caused unprecedented impacts to people's daily life around the world. Various measures and policies such as lockdown and social-distancing are implemented by governments to combat the disease during the pandemic period. These measures and policies as well as virus itself may cause different mental health issues to people such as depression, anxiety, sadness, etc. In this paper, we exploit the massive text data posted by Twitter users to analyse the sentiment dynamics of people living in the state of New South Wales (NSW) in Australia during the pandemic period. Different from the existing work that mostly focuses on the country-level and static sentiment analysis, we analyse the sentiment dynamics at the fine-grained local government areas (LGAs). Based on the analysis of around 94 million tweets that posted by around 183 thousand users located at different LGAs in NSW in 5 months, we found that people in NSW showed an overall positive sentimental polarity and the COVID-19 pandemic decreased the overall positive sentimental polarity during the pandemic period. The fine-grained analysis of sentiment in LGAs found that despite the dominant positive sentiment most of days during the study period, some LGAs experienced significant sentiment changes from positive to negative. This study also analysed the sentimental dynamics delivered by the hot topics in Twitter such as government policies (e.g. the Australia's JobKeeper program, lockdown, social-distancing) as well as the focused social events (e.g. the Ruby Princess Cruise). The results showed that the policies and events did affect people's overall sentiment, and they affected people's overall sentiment differently at different stages.