This work introduces a microwave-based system able to detect tumours in breast phantoms in a non-invasive way. The data acquisition system is composed of a hardware system which involves high-frequency components (antennas, switches and cables), a microcontroller, a vector network analyser used as measurement instrument and a computer devoted to the control and automation of the operation of the system. Concerning the software system, the computer runs a Python script which is in charge of mastering and automatising all the required stages for the data acquisition, from initialisation of the hardware system to performing and saving the measurements. We also report on the design of the high-performance broadband antenna used to carry out the measurements, as well as on the algorithm employed to build the final medical images, based on an adapted version of the so-called Improved Delay-and-Sum (IDAS) algorithm improved by a Hamming window filter and averaging preprocessing. The calibration and start-up of the system are also described. The experimental validation includes the use of different tumour models with different dielectric properties inside the breast phantom. The results show promising tumour detection capabilities, even when there is low dielectric contrast between the tumoural and healthy tissues, as is the usual case for dense breasts in young women.