  We study generalised Heegner cycles, originally introduced by
Bertolini-Darmon-Prasanna for modular curves, in the context of Mumford curves.
The main result of this paper relates generalized Heegner cycles with the two
variable anticyclotomic $p$-adic $L$-function attached to a Coleman family
$f_\infty$ and an imaginary quadratic field $K$. Our generalised Heegner cycles
allow us to study the restriction of this function to non-central critical
lines. The main result expresses the derivative along the weight variable of
this anticyclotomic $p$-adic $L$-function restricted to non necessarily central
critical lines as a combination of the image of generalized Heegner cycles
under a $p$-adic Abel-Jacobi map. In studying generalised Heegner cycles in the
context of Mumford curves, we also obtain an extension of a result of Masdeu
for the (one variable) anticyclotomic $p$-adic $L$-function of a modular form
$f$ and an imaginary quadratic field $K$ at non-central critical integers.

Number Theory