In a cell-free massive multiple-input multiple-output (CF-mMIMO) network, multiple access points (APs) actively cooperate to serve users' equipment (UEs). We consider how the random access (RA) problem can be addressed by such a network under the occurrence of pilot collisions. To find a solution, we embrace the user-centric perspective, which basically dictates that only a preferred set of APs needs to serve a UE. Due to the success of the strongest-user collision resolution (SUCRe) protocol for cellular (Ce) mMIMO, we extend it by considering the new setting. Besides, we establish that the user-centric perspective naturally equips a CF network with robust fundamentals for resolving collisions. We refer to this foundation as spatial separability, which enables multiple colliding UEs to access the network simultaneously. We then propose two novel RA protocols for CF-mMIMO: i) the baseline cell-free (BCF) that resolves collisions with the concept of spatial separability alone, and ii) the cell-free SUCRe (CF-SUCRe) that combines SUCRe and spatial separability principle to resolve collisions. We evaluate our proposed RA protocols against the Ce-SUCRe. Respectively, the BCF and CF-SUCRe can support 7x and 4x more UEs' access on average compared to the Ce-SUCRe with an average energy efficiency gain based on total power consumed (TPC) by the network per access attempt of 52x and 340x. Among our procedures, even with a higher overhead, the CF-SUCRe is superior to BCF regarding TPC per access attempt. This is because the combination of methods for collision resolution allows many APs to be disconnected from the RA process without sacrificing much the performance. Finally, our numerical results presented in this work can be reproduced using the code package available on: github.com/victorcroisfelt/cf-ra-spatial-separability.