Voluntary shelter-in-place directives and lockdowns are the main non-pharmaceutical interventions that governments around the globe have used to contain the Covid-19 pandemic. In this paper we study the impact of such interventions in the capital of a developing country, Santiago, Chile, that exhibits large socioeconomic inequality. A distinctive feature of our study is that we use granular geolocated mobile phone data to construct mobility measures that capture (1) shelter-in-place behavior, and (2) trips within the city to destinations with potentially different risk profiles. Using panel data linear regression models we first show that the impact of social distancing measures and lockdowns on mobility is highly heterogeneous and dependent on socioeconomic levels. More specifically, our estimates indicate that while zones of high socioeconomic levels can exhibit reductions in mobility of around 50% to 90% depending on the specific mobility metric used, these reductions are only 20% to 50% for lower-income communities. The large reductions in higher-income communities are significantly driven by voluntary shelter-in-place behavior. Second, also using panel data methods we show that our mobility measures are important predictors of infections: a 10% increase in mobility correlates with approximately a 5% increase in the rate of infection. Our results suggest that mobility is an important factor explaining differences in infections rates between high and low incomes areas within the city. Further, they confirm the challenges of reducing mobility in lower-income communities, where people generate their income from their daily work. To be effective, shelter-in-place restrictions in municipalities of low socioeconomic levels may need to be complemented by other supporting measures that enable their inhabitants to increase compliance.