Processing-using-DRAM (PuD) is a promising paradigm for alleviating the data movement bottleneck using DRAM's massive internal parallelism and bandwidth to execute very wide operations. Performing a PuD operation involves activating multiple DRAM rows in quick succession or simultaneously, i.e., multiple-row activation. Multiple-row activation is fundamentally different from conventional memory access patterns that activate one DRAM row at a time. However, repeatedly activating even one DRAM row (e.g., RowHammer) can induce bitflips in unaccessed DRAM rows because modern DRAM is subject to read disturbance. Unfortunately, no prior work investigates the effects of multiple-row activation on DRAM read disturbance. In this paper, we present the first characterization study of read disturbance effects of multiple-row activation-based PuD (which we call PuDHammer) using 316 real DDR4 DRAM chips from four major DRAM manufacturers. Our detailed characterization show that 1) PuDHammer significantly exacerbates the read disturbance vulnerability, causing up to 158.58x reduction in the minimum hammer count required to induce the first bitflip ($HC_{first}$), compared to RowHammer, 2) PuDHammer is affected by various operational conditions and parameters, 3) combining RowHammer with PuDHammer is more effective than using RowHammer alone to induce read disturbance error, e.g., doing so reduces $HC_{first}$ by 1.66x on average, and 4) PuDHammer bypasses an in-DRAM RowHammer mitigation mechanism (Target Row Refresh) and induces more bitflips than RowHammer. To develop future robust PuD-enabled systems in the presence of PuDHammer, we 1) develop three countermeasures and 2) adapt and evaluate the state-of-the-art RowHammer mitigation standardized by industry, called Per Row Activation Counting (PRAC). We show that the adapted PRAC incurs large performance overheads (48.26%, on average).