We present the analysis of four first overtone RR Lyrae stars observed with the Kepler space telescope, based on data obtained over nearly 2.5 yr. All four stars are found to be multiperiodic. The strongest secondary mode with frequency f... has an amplitude of a few mmag, 20-45 times lower than the main radial mode with frequency f... The two oscillations have a period ratio of P.../P... = 0.612-0.632 that cannot be reproduced by any two radial modes. Thus, the secondary mode is non-radial. Modes yielding similar period ratios have also recently been discovered in other variables of the RRc and RRd types. These objects form a homogenous group and constitute a new class of multimode RR Lyrae pulsators, analogous to a similar class of multimode classical Cepheids in the Magellanic Clouds. Because a secondary mode with P.../P... ~ 0.61 is found in almost every RRc and RRd star observed from space, this form of multiperiodicity must be common. In all four Kepler RRc stars studied, we find subharmonics of f... at ~1/2f... and at ~3/2f... This is a signature of period doubling of the secondary oscillation, and is the first detection of period doubling in RRc stars. The amplitudes and phases of f... and its subharmonics are variable on a time-scale of 10-200 d. The dominant radial mode also shows variations on the same time-scale, but with much smaller amplitude. In three Kepler RRc stars we detect additional periodicities, with amplitudes below 1 mmag, that must correspond to non-radial g-modes. Such modes never before have been observed in RR Lyrae variables. (ProQuest: ... denotes formulae/symbols omitted.)