α-RuCl3 has attracted enormous attention since it has been proposed as a prime candidate to study fractionalized magnetic excitations akin to Kitaev's honeycomb-lattice spin liquid. We have performed a detailed specific-heat investigation at temperatures down to 0.4 K in applied magnetic fields up to 9 T for fields parallel to the ab plane. We find a suppression of the zero-field antiferromagnetic order, together with an increase of the low-temperature specific heat, with increasing field up to μ0Hc ≈ 6.9 T. Above Hc, the magnetic contribution to the low-temperature specific heat is strongly suppressed, implying the opening of a spin-excitation gap. Our data point toward a field-induced quantum critical point at Hc; this is supported by universal scaling behavior near Hc. Remarkably, the data also reveal the existence of a small characteristic energy scale well below 1 meV, above which the excitation spectrum changes qualitatively. We relate the data to theoretical calculations based on a J1-K1-Γ1-J3 honeycomb model.