  Let $T$ be a polynomially bounded o-minimal theory extending the theory of
real closed ordered fields. Let $K$ be a model of $T$ equipped with a
$T$-convex valuation ring and a $T$-derivation. If this derivation is
continuous with respect to the valuation topology, then we call $K$ a
$T$-convex $T$-differential field. We show that every $T$-convex
$T$-differential field has an immediate strict $T$-convex $T$-differential
field extension which is spherically complete. In some important cases, the
assumption of polynomial boundedness can be relaxed to power boundedness.
