  In the paper [On superspecial abelian surfaces over finite fields II. J.
Math. Soc. Japan, 72(1):303--331, 2020], Tse-Chung Yang and the first two
current authors computed explicitly the number $\lvert
\mathrm{SSp}_2(\mathbb{F}_q)\rvert$ of isomorphism classes of superspecial
abelian surfaces over an arbitrary finite field $\mathbb{F}_q$ of even degree
over the prime field $\mathbb{F}_p$. There it was assumed that certain
commutative $\mathbb{Z}_p$-orders satisfy an \'etale condition that excludes
the primes $p=2, 3, 5$. We treat these remaining primes in the present paper,
where the computations are more involved because of the ramifications. This
completes the calculation of $\lvert \mathrm{SSp}_2(\mathbb{F}_q)\rvert$ in the
even degree case. The odd degree case was previous treated by Tse-Chung Yang
and the first two current authors in [On superspecial abelian surfaces over
finite fields. Doc. Math., 21:1607--1643, 2016]. Along the proof of our main
theorem, we give the classification of lattices over local quaternion Bass
orders, which is a new input to our previous works.
