In a cell-free massive multiple-input multiple-output (mMIMO) network, multiple access points (APs) actively cooperate to serve users' equipment (UEs). We consider how the problem of random access (RA) to pilots 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 subset of APs needs to service a UE. Due to the success of the strongest-user collision resolution (SUCRe) protocol for cellular mMIMO, we propose an extension of SUCRe considering the new setting. During the extension process, we observe that the user-centric perspective naturally equips a cell-free network with a method for resolving collisions. We refer to this method as spatial separability, which comes from the macro-diversity brought about by the additional AP dimension. We then propose two novel RA protocols for cell-free mMIMO: i) the baseline cell-free (BCF) that resolves collisions with the concept of separability alone and ii) the cell-free SUCRe (CF-SUCRe) that combines SUCRe and spatial separability to resolve collisions. We evaluate our proposed RA protocols against the cellular SUCRe (Ce-SUCRe). The BCF and CF-SUCRe perform 6x to 2x better on average compared to the Ce-SUCRe with an average energy efficiency gain of 45x to 125x, respectively. Among our methods, even with a higher overhead, the CF-SUCRe is superior to BCF because the combination of methods for collision resolution allows many APs to be disconnected from RA without sacrificing much performance.