  We present the discovery of a new, peculiar form of double-periodic pulsation
in RR Lyrae stars. In four, long-period ($P>0.6$ d) stars observed by the
Optical Gravitational Lensing Experiment, and classified as fundamental mode
pulsators (RRab), we detect additional, low-amplitude variability, with period
shorter than fundamental mode period. The period ratios fall in a range similar
to double-mode fundamental and first overtone RR Lyrae stars (RRd), with the
exception of one star, in which the period ratio is significantly lower and
nearly exactly equals 0.7. Although period ratios are fairly different for the
four stars, the light curve shapes corresponding to the dominant, fundamental
mode are very similar. The peak-to-peak amplitudes and amplitude ratios
(Fourier parameters $R_{21}$ and $R_{31}$) are among the highest observed in
RRab stars of similar period, while Fourier phases ($\varphi_{21}$ and
$\varphi_{31}$) are among the lowest observed in RRab stars.
  If the additional variability is interpreted as due to radial first overtone,
then, the four stars are the most extreme RRd variables of the longest
pulsation periods known. Indeed, the observed period ratios can be well
modelled with high metallicity pulsation models. However, at such long
pulsation periods, first overtone is typically damped.
  Five other candidates, with weak signature of additional variability, sharing
the same characteristics, were also detected and are briefly discussed.
