  We study the dynamics of the nonlinear $q$-voter model with inflexible
zealots in a finite well-mixed population. In this system, each individual
supports one of two parties and is either a susceptible voter or an inflexible
zealot. At each time step, a susceptible adopts the opinion of a neighbor if
this belongs to a group of $q\geq 2$ neighbors all in the same state, whereas
inflexible zealots never change their opinion. In the presence of zealots of
both parties the model is characterized by a fluctuating stationary state and,
below a zealotry density threshold, the distribution of opinions is bimodal.
After a characteristic time, most susceptibles become supporters of the party
having more zealots and the opinion distribution is asymmetric. When the number
of zealots of both parties is the same, the opinion distribution is symmetric
and, in the long run, susceptibles endlessly swing from the state where they
all support one party to the opposite state. Above the zealotry density
threshold, when there is an unequal number of zealots of each type, the
probability distribution is single-peaked and non-Gaussian. These properties
are investigated analytically and with stochastic simulations. We also study
the mean time to reach a consensus when zealots support only one party.
