Sum-rank Hamming codes are introduced in this work. They are essentially defined as the longest codes (thus of highest information rate) with minimum sum-rank distance at least $ 3 $ (thus one-error-correcting) for a fixed redundancy $ r $, base-field size $ q $ and field-extension degree $ m $ (i.e., number of matrix rows). General upper bounds on their code length, number of shots or sublengths and average sublength are obtained based on such parameters. When the field-extension degree is $ 1 $, it is shown that sum-rank isometry classes of sum-rank Hamming codes are in bijective correspondence with maximal-size partial spreads. In that case, it is also shown that sum-rank Hamming codes are perfect codes for the sum-rank metric. Also in that case, estimates on the parameters (lengths and number of shots) of sum-rank Hamming codes are given, together with an efficient syndrome decoding algorithm. Duals of sum-rank Hamming codes, called sum-rank simplex codes, are then introduced. Bounds on the minimum sum-rank distance of sum-rank simplex codes are given based on known bounds on the size of partial spreads. As applications, sum-rank Hamming codes are proposed for error correction in multishot matrix-multiplicative channels and to construct locally repairable codes over small fields, including binary.