  It is widely accepted that dark matter contributes about a quarter of the
critical mass-energy density in our Universe. The nature of dark matter is
currently unknown, with the mass of possible constituents spanning nearly one
hundred orders of magnitude. The ultralight scalar field dark matter,
consisting of extremely light bosons with $m \sim 10^{-22}$ eV and often called
"fuzzy" dark matter, provides intriguing solutions to some challenges at
sub-Galactic scales for the standard cold dark matter model. As shown by
Khmelnitsky and Rubakov, such a scalar field in the Galaxy would produce an
oscillating gravitational potential with nanohertz frequencies, resulting in
periodic variations in the times of arrival of radio pulses from pulsars. The
Parkes Pulsar Timing Array (PPTA) has been monitoring 20 millisecond pulsars at
two to three weeks intervals for more than a decade. In addition to the
detection of nanohertz gravitational waves, PPTA offers the opportunity for
direct searches for fuzzy dark matter in an astrophysically feasible range of
masses. We analyze the latest PPTA data set which includes timing observations
for 26 pulsars made between 2004 and 2016. We perform a search in this data set
for evidence of ultralight dark matter in the Galaxy using Bayesian and
Frequentist methods. No statistically significant detection has been made. We
therefore place upper limits on the local dark matter density. Our limits,
improving on previous searches by a factor of two to five, constrain the dark
matter density of ultralight bosons with $m \leq 10^{-23}$ eV to be below
$6\,\text{GeV}\,\text{cm}^{-3}$ with 95\% confidence in the Earth neighborhood.
Finally, we discuss the prospect of probing the astrophysically favored mass
range $m \gtrsim 10^{-22}$ eV with next-generation pulsar timing facilities.
