  Let $E/\mathbf{Q}$ be an elliptic curve and let $p$ be an odd prime of good
reduction for $E$. Let $K$ be an imaginary quadratic field satisfying the
classical Heegner hypothesis and in which $p$ splits. In a previous work,
Agboola--Castella formulated an analogue of the Birch--Swinnerton-Dyer
conjecture for the $p$-adic $L$-function $L_{\mathfrak{p}}^{\rm BDP}$ of
Bertolini--Darmon--Prasanna attached to $E/K$, assuming the prime $p$ to be
ordinary for $E$. The goal of this paper is two-fold:
  (1) We formulate a $p$-adic BSD conjecture for $L_{\mathfrak{p}}^{\rm BDP}$
for all odd primes $p$ of good reduction.
  (2) For an algebraic analogue $F_{\overline{\mathfrak{p}}}^{\rm BDP}$ of
$L_{\mathfrak{p}}^{\rm BDP}$, we show that the ``leading coefficient'' part of
our conjecture holds, and that the ``order of vanishing'' part follows from the
expected ``maximal non-degeneracy'' of an anticyclotomic $p$-adic height.
  In particular, when the Iwasawa--Greenberg Main Conjecture
$(F_{\overline{\mathfrak{p}}}^{\rm BDP})=(L_{\mathfrak{p}}^{\rm BDP})$ is
known, our results determine the leading coefficient of $L_{\mathfrak{p}}^{\rm
BDP}$ at $T=0$ up to a $p$-adic unit. Moreover, by adapting the approach of
Burungale--Castella--Kim in the $p$-ordinary case, we prove the main conjecture
for supersingular primes $p$ under mild hypotheses.
