A continuous phase transition driven to zero temperature by a non-thermal parameter, such as pressure, terminates in a quantum critical point (QCP). At present, two main theoretical approaches are available for antiferromagnetic QCPs in heavy-fermion systems. The conventional one is the quantum generalization of finite-temperature phase transitions, which reproduces the physical properties in many cases^,,,,. More recent unconventional models incorporate a breakdown of the Kondo effect, giving rise to a Fermi-surface reconstruction^,,— YbRh₂Si₂ is a prototype of this category^,,,. In YbRh₂Si₂, the antiferromagnetic transition temperature merges with the Kondo breakdown at the QCP. Here, we study the evolution of the quantum criticality in YbRh₂Si₂ under chemical pressure. Surprisingly, for positive pressure we find the signature of the Kondo breakdown within the magnetically ordered phase, whereas negative pressure induces their separation, leaving an intermediate spin-liquid-type ground state over an extended range. This behaviour suggests a new quantum phase arising from the interplay of the Kondo breakdown and the antiferromagnetic QCP.