  The characteristics of Fe-based superconductors are manifested in their
electronic, magnetic properties, and pairing symmetry of the Cooper pair, but
the latter remain to be explored. Usually in these materials, superconductivity
coexists and competes with magnetic order, giving unconventional pairing
mechanisms. We report on the results of the bulk magnetization measurements in
the superconducting state and the low-temperature specific heat down to 0.4 K
for BaFe$_{2-x}$Ni$_{x}$As$_{2}$ single crystals. The {electronic} specific
heat displays a pronounced anomaly at the superconducting transition
temperature and a small residual part {at low temperatures in the
superconducting state}. The normal-state Sommerfeld coefficient increases with
Ni doping for $x$ = 0.092, 0.096, and 0.10, which illustrates the competition
between magnetism and superconductivity. Our analysis of the temperature
dependence of the superconducting-state specific heat and the London
penetration depth provides strong evidence for a two-band $s$-wave order
parameter. Further, the data of the London penetration depth calculated from
the lower critical field follow an exponential temperature dependence,
characteristic of a fully gapped superconductor. These observations clearly
show that the superconducting gap in the nearly optimally doped compounds is
nodeless.
