We consider unsourced random access (uRA) in user-centric cell-free (CF) wireless networks, where random access users send codewords from a common codebook during specifically dedicated random access channel (RACH) slots. The system is conceptually similar to the so-called 2-step RACH currently discussed in 3GPP standardization. In order to cope with the distributed and CF nature of the network, we propose to partition the network coverage area into zones (referred to as ''locations'') and assign an uRA codebook to each location, such that users in a certain location make use of the associated codebook. The centralized uRA decoder makes use of the multisource AMP algorithm recently proposed by the authors. This yields at once the list of active uRA codewords, an estimate of the corresponding channel vectors, and an estimate of the active users' position. We show excellent performance of this approach and perfect agreement with the rigorous theoretical ''state evolution'' analysis. We also show that the proposed ''location-based'' partitioned codebook approach significantly outperforms a baseline system with a single non-partitioned uRA codebook.