It is shown that a set-valued mapping
T
∗
T^{\ast }
of a hyperconvex metric space
M
M
which takes values in the space of nonempty externally hyperconvex subsets of
M
M
always has a lipschitzian single valued selection
T
T
which satisfies
d
(
T
(
x
)
,
T
(
y
)
)
≤
d
H
(
T
∗
(
x
)
,
T
∗
(
y
)
)
d(T(x),T(y))\leq d_{H}(T^{\ast }(x),T^{\ast }(y))
for all
x
,
y
∈
M
x,y\in M
. (Here
d
H
d_{H}
denotes the usual Hausdorff distance.) This fact is used to show that the space of all bounded
λ
\lambda
-lipschitzian self-mappings of
M
M
is itself hyperconvex. Several related results are also obtained.