We present low-temperature thermopower results, S(T), on the heavy-fermion compound YbRh2Si2 in the vicinity of its field-induced quantum critical point (QCP). At B=0, a logarithmic increase of -S(T)/T between 1 and 0.1 K reveals strong non-Fermi-liquid behavior. A pronounced downturn of -S(T)/T below T{max}=0.1 K and a sign change from negative to positive S(T) values at T{0} approximately 30 mK are observed on the low-field side of the Kondo breakdown crossover line T{*}(B). In the field-induced, heavy Landau-Fermi-liquid regime, S(T)/T assumes constant, negative values below T{LFL}. A pronounced crossover in the -S(B)/T isotherms at T{*}(B) sharpens with decreasing T and seems to evolve toward a steplike function for T-->0. This is attributed to an abrupt change of the Fermi volume upon crossing the unconventional QCP of YbRh2Si2.