  YBa$_2$Cu$_3$O$_{7-\delta}$ (YBCO) has favorable macroscopic superconducting
properties of $T_\mathrm{c}$ up to 93 K and $H_{c2}$ up to 150 T. However, its
nanoscale electronic structure remains mysterious because bulk-like electronic
properties are not preserved near the surface of cleaved samples for easy
access by local or surface-sensitive probes. It has been hypothesized that
Ca-doping at the Y site could induce an alternate cleavage plane that mitigates
this issue. We use scanning tunneling microscopy (STM) to study both Ca-free
and 10% Ca-doped YBCO, and find that the Ca-doped samples do indeed cleave on
an alternate plane, yielding a spatially-disordered partial (Y,Ca) layer. Our
density functional theory calculations support the increased likelihood of this
new cleavage plane in Ca-doped YBCO. On this surface, we image a
superconducting gap with average value 24 $\pm$ 3 meV and characteristic length
scale 1-2 nm, similar to Bi-based high-$T_\mathrm{c}$ cuprates, but the first
map of gap inhomogeneity in the YBCO family.
