During chronic infection, HIV-1 engages in a rapid coevolutionary arms race with the host’s adaptive immune system. While it is clear that HIV exerts strong selection on the adaptive immune system, the characteristics of the somatic evolution that shape the immune response are still unknown. Traditional population genetics methods fail to distinguish chronic immune response from healthy repertoire evolution. Here, we infer the evolutionary modes of B-cell repertoires and identify complex dynamics with a constant production of better B-cell receptor (BCR) mutants that compete, maintaining large clonal diversity and potentially slowing down adaptation. A substantial fraction of mutations that rise to high frequencies in pathogen-engaging CDRs of BCRs are beneficial, in contrast to many such changes in structurally relevant frameworks that are deleterious and circulate by hitchhiking. We identify a pattern where BCRs in patients who experience larger viral expansions undergo stronger selection with a rapid turnover of beneficial mutations due to clonal interference in their CDR3 regions. Using population genetics modeling, we show that the extinction of these beneficial mutations can be attributed to the rise of competing beneficial alleles and clonal interference. The picture is of a dynamic repertoire, where better clones may be outcompeted by new mutants before they fix.