The use of shape memory alloys for micro-actuators constitutes a field of application in which copper–aluminum-based alloys find their usefulness because they can reach higher activation temperatures and are easier to produce than titanium-based alloys, particularly by the method proposed in this work. SMA tapes are a two-dimensional structure that offers many design options such as stamping, punching, and deep drawing, but they are also suitable for laser cutting, engraving, stamping, and EDM machining. This work has been made to study the manufacture of copper-based shape memory alloys (SMAs) using the cold co-rolling process also called the cold-roll bonding (CRB) process. In this process, a thin metal sandwich can be produced with a rolling machine. This sandwich consists of layers of CuNiBe master alloy and Al. During the rolling phase, the sandwich has no shape memory effect (SME) or superelastic effect (SE), so thin strips can be easily produced. After the rolling phase, the sandwich is subjected to a complex heat treatment to gain the SME. To validate this process to produce Cu-based SMAs, several alloys with different CuAlNiBe compositions have been tested. The SMAs obtained were characterized by optical microscopy, scanning electron microscopy (SEM) and X-ray diffraction (XRD) techniques. The martensitic transformation was studied by Differential Scanning Calorimetry (DSC) and SME and SE were studied by three-point bending tests. This work shows that the CRB is a good process for making a wide variety of Cu-based SMA ribbons.