Reconfigurable intelligent surfaces (RISs) have emerged as a spectrum- and energy-efficient technology to enhance the coverage of wireless communications within the upcoming 6G networks. Recently, novel extensions of this technology, referred to as multi-sector beyond diagonal RIS (BD-RIS), have been proposed, where the configurable elements are divided into $L$ sectors $(L \geq 2)$ and arranged as a polygon prism, with each sector covering $1/L$ space. This paper presents a performance analysis of a multi-user communication system assisted by a multi-sector BD-RIS operating in time-switching (TS) mode. Specifically, we derive closed-form expressions for the moment-generating function (MGF), probability density function (PDF), and cumulative density function (CDF) of the signal-to-noise ratio (SNR) per user. Furthermore, closed-form expressions for the outage probability, achievable spectral and energy efficiency, symbol error probability, and diversity order for the proposed system model are derived. Moreover, a comparison is performed with the simultaneously transmitting and reflecting (STAR)-RISs, a special case of multi-sector BD-RIS with two sectors. Our analysis shows that for a fixed number of elements, increasing the sectors improves outage performance at the expense of reduced diversity order compared to STAR-RIS. This trade-off is influenced by the Rician factors of the cascaded channel and the number of configurable elements per sector. However, this superiority in slope is observed at outage probability values below $10^{-5}$, which remains below practical operating ranges of communication systems. Additionally, simulations are provided to validate the accuracy of our theoretical analyses showing a notable $182\%$ increase in spectral efficiency and a $238\%$ increase in energy efficiency when transitioning from a 2-sector to a 6-sector configuration.