We present observations of transverse oscillations in slender Ca ii H fibrils ( SCFs ) in the lower solar chromosphere . We use a 1 hr long time series of high- ( spatial and temporal- ) resolution seeing-free observations in a 1.1 Å wide passband covering the line core of Ca ii H 3969 Å from the second flight of the Sunrise balloon-borne solar observatory . The entire field of view , spanning the polarity inversion line of an active region close to the solar disk center , is covered with bright , thin , and very dynamic fine structures . Our analysis reveals the prevalence of transverse waves in SCFs with median amplitudes and periods on the order of 2.4 \pm 0.8 km/s and 83 \pm 29 s , respectively ( with standard deviations given as uncertainties ) . We find that the transverse waves often propagate along ( parts of ) the SCFs with median phase speeds of 9 \pm 14 km/s . While the propagation is only in one direction along the axis in some of the SCFs , propagating waves in both directions , as well as standing waves are also observed . The transverse oscillations are likely Alfvénic and are thought to be representative of magnetohydrodynamic kink waves . The wave propagation suggests that the rapid high-frequency transverse waves , often produced in the lower photosphere , can penetrate into the chromosphere with an estimated energy flux of \approx 15 kW/m ^ { 2 } . Characteristics of these waves differ from those reported for other fibrillar structures , which , however , were observed mainly in the upper solar chromosphere .