We present observations of transverse oscillations in slender Ca IIH fibrils (SCFs) in the lower solar chromosphere. We use a 1 hr long time series of high-(spatial and temporal-) resolutionseeing-free observations in a 1.1 Å wide passband covering the line core of Ca IIH 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±0.8 km s −1 and 83±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±14 km s −1. 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 rapidhigh-frequency transverse waves,often produced in the lower photosphere, can penetrate into the chromospherewith an estimated energy flux of ≈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.