We propose an irregular repetition slotted ALOHA (IRSA) based random-access protocol for the binary adder channel (BAC). The BAC captures important physical-layer concepts, such as packet generation, per-slot decoding, and information rate, which are neglected in the commonly considered collision channel model. We divide a frame into slots and let users generate a packet, to be transmitted over a slot, from a given codebook. In a state-of-the-art scheme proposed by Paolini et al. (2022), the codebook is constructed as the parity-check matrix of a BCH code. Here, we construct the codebook from independent and identically distributed binary symbols to obtain a random-coding achievability bound. Our per-slot decoder progressively discards incompatible codewords from a list of candidate codewords, and can be improved by shrinking this list across iterations. In a regime of practical interests, our scheme can resolve more colliding users in a slot and thus achieves a higher average sum rate than the scheme in Paolini et al. (2022).