  A topological space $X$ is said to be {\em $Y$-rigid} if any continuous map
$f:X\rightarrow Y$ is constant. In this paper we construct a number of examples
of regular countably compact $\mathbb R$-rigid spaces with additional
properties like separability and first countability. This way we answer several
questions of Tzannes, Banakh, Ravsky, as well as get a consistent example of
$\mathbb R$-rigid Nyikos space. Also, we show that it is consistent with ZFC
that for every cardinal $\kappa<\mathfrak c$ there exists a regular separable
countably compact space $X$ which is $Y$-rigid with respect to any $T_1$ space
$Y$ of pseudocharacter $\leq\kappa$.
