As conventional circuits based on field-effect transistors are approaching their physical limits due to quantum phenomena, semi-floating gate transistors have emerged as an alternative ultrafast and silicon-compatible technology. Here, we show a quasi-non-volatile memory featuring a semi-floating gate architecture with band-engineered van der Waals heterostructures. This two-dimensional semi-floating gate memory demonstrates 156 times longer refresh time with respect to that of dynamic random access memory and ultrahigh-speed writing operations on nanosecond timescales. The semi-floating gate architecture greatly enhances the writing operation performance and is approximately 106 times faster than other memories based on two-dimensional materials. The demonstrated characteristics suggest that the quasi-non-volatile memory has the potential to bridge the gap between volatile and non-volatile memory technologies and decrease the power consumption required for frequent refresh operations, enabling a high-speed and low-power random access memory.