DNA sequencing data continue to progress toward longer reads with increasingly lower sequencing error rates. We focus on the critical problem of mapping, or aligning, low-divergence sequences from long reads (e.g., Pacific Biosciences [PacBio] HiFi) to a reference genome, which poses challenges in terms of accuracy and computational resources when using cutting-edge read mapping approaches that are designed for all types of alignments. A natural idea would be to optimize efficiency with longer seeds to reduce the probability of extraneous matches; however, contiguous exact seeds quickly reach a sensitivity limit. We introduce mapquik, a novel strategy that creates accurate longer seeds by anchoring alignments through matches of k consecutively sampled minimizers (k-min-mers) and only indexing k-min-mers that occur once in the reference genome, thereby unlocking ultrafast mapping while retaining high sensitivity. We show that mapquik significantly accelerates the seeding and chaining steps-fundamental bottlenecks to read mapping-for both the human and maize genomes with >96% sensitivity and near-perfect specificity. On the human genome, for both real and simulated reads, mapquik achieves a 37x speedup over the state-of-the-art tool minimap2, and on the maize genome, mapquik achieves a 410x speedup over minimap2, making mapquik the fastest mapper to date. These accelerations are enabled from not only minimizer-space seeding but also a novel heuristic O(n) pseudochaining algorithm, which improves upon the long-standing O(nlogn) bound. Minimizer-space computation builds the foundation for achieving real-time analysis of long-read sequencing data.