  We introduce the strong Gelfand-Phillips property for locally convex spaces
and give several characterizations of this property. We characterize the strong
Gelfand-Phillips property among locally convex spaces admitting a stronger
Banach space topology. If $C_{\mathcal T}(X)$ is a space of continuous
functions on a Tychonoff space $X$, endowed with a locally convex topology
$\mathcal T$ between the pointwise topology and the compact-open topology,
then: (a) the space $C_{\mathcal T}(X)$ has the strong Gelfand-Phillips
property iff $X$ contains a compact countable subspace $K\subseteq X$ of finite
scattered height such that for every functionally bounded set $B\subseteq X$
the complement $B\setminus K$ is finite, (b) the subspace $C^b_{\mathcal T}(X)$
of $C_{\mathcal T}(X)$ consisting of all bounded functions on $X$ has the
strong Gelfand-Phillips property iff $X$ is a compact countable space of finite
scattered height.
