  We show that there exists a de Branges-Rovnyak space $\mathcal{H}(b)$ on the
unit disk containing a function $f$ with the following property: even though
$f$ can be approximated by polynomials in $\mathcal{H}(b)$, neither the Taylor
partial sums of $f$ nor their Ces\`aro, Abel, Borel or logarithmic means
converge to $f$ in $\mathcal{H}(b)$.
  A key tool is a new abstract result showing that, if one regular summability
method includes another for scalar sequences, then it automatically does so for
certain Banach-space-valued sequences too.
