  It is known that there exist functions in certain de Branges--Rovnyak spaces
whose Taylor series diverge in norm, even though polynomials are dense in the
space. This is often proved by showing that the sequence of Taylor partial sums
is unbounded in norm. In this note we show that it can even happen that the
Taylor partial sums tend to infinity in norm. We also establish similar results
for lower-triangular summability methods such as the Ces\`aro means.
