Liquid dispenser systems, pumps for use in liquid dispenser systems, and disposable/replaceable liquid container units for use in liquid dispenser systems are disclosed. A refill unit for refilling a liquid dispenser system comprises a container for holding a supply of liquid and a pump connected to the container. The pump comprises a fitment portion configured for attachment to a liquid container, and a pump portion which comprises an upper valve member and a lower valve member disposed along a common longitudinal axis, such that the fitment portion and the pump portion are formed as one integral piece.