Fractionalized excitations are of considerable interest in recent condensed-matter physics. Fractionalization of the spin degrees of freedom into localized and itinerant Majorana fermions is predicted for the Kitaev spin liquid, an exactly solvable model with bond-dependent interactions on a two-dimensional honeycomb lattice. As a function of temperature, theory predicts a characteristic two-peak structure of the heat capacity as a fingerprint of these excitations. Here we report on detailed heat-capacity experiments as a function of temperature and magnetic field in high-quality single crystals of α-RuCl3. We undertook considerable efforts to determine the exact phonon background. We measured single-crystalline RhCl3 as a nonmagnetic reference and performed ab initio calculations of the phonon density of states for both compounds. These ab initio calculations document that the intrinsic phonon contribution to the heat capacity cannot be obtained by a simple rescaling of the nonmagnetic reference using differences in the atomic masses. Sizable renormalization is required even for nonmagnetic RhCl3 with its minute difference from the title compound. In α-RuCl3 in zero magnetic field, excess heat capacity exists at temperatures well above the onset of magnetic order. In external magnetic fields far beyond quantum criticality, when long-range magnetic order is fully suppressed, the excess heat capacity exhibits the characteristic two-peak structure. In zero field, the lower peak just appears at temperatures around the onset of magnetic order and seems to be connected with canonical spin degrees of freedom. At higher fields, beyond the critical field, this peak is shifted to 10 K. The high-temperature peak located around 70 K is hardly influenced by external magnetic fields, carries the predicted amount of entropy R/2ln2, and may resemble remnants of Kitaev physics.