Amorphous water ice comes in at least three distinct structural forms, all lacking long-range crystalline order. High-density amorphous ice (HDA) was first produced by compressing ice I to 11 kilobar at temperatures below 130 kelvin, and the process was described as thermodynamic melting(1), implying that HDA is a glassy state of water. This concept, and the ability to transform HDA reversibly into low-density amorphous ice, inspired the two-liquid water model, which relates the amorphous phases to two liquid waters in the deeply supercooled regime (below 228 kelvin) to explain many of the anomalies of water(2) (such as density and heat capacity anomalies). However, HDA formation has also been ascribed3 to a mechanical instability causing structural collapse and associated with kinetics too sluggish for recrystallization to occur. This interpretation is supported by simulations(3), analogy with a structurally similar system(4), and the observation of lattice-vibration softening as ice is compressed(5,6). It also agrees with recent observations of ice compression at higher temperatures-in the 'no man's land' regime, between 145 and 200 kelvin, where kinetics are faster-resulting in crystalline phases(7,8). Here we further probe the role of kinetics and show that, if carried out slowly, compression of ice I even at 100 kelvin (a region in which HDA typically forms) gives proton-ordered, but non-interpenetrating, ice IX', then proton-ordered and interpenetrating ice XV', and finally ice VIII'. By contrast, fast compression yields HDA but no ice IX, and direct transformation of ice I to ice XV' is structurally inhibited. These observations suggest that HDA formation is a consequence of a kinetically arrested transformation between low-density ice I and high-density ice XV' and challenge theories that connect amorphous ice to supercooled liquid water.