The answer to the question is yes for the upper bound (albeit with different constants in the gaussian factor), and no for the lower bound.

For the upper bound, and for any domain $\Omega \subset M$, by domain monotonicity of the heat kernel (e.g. cf. [1, Exercise 7.40]), you have

$$p_\Omega(t,x,y) \leq p(t,x,y), \qquad \forall x,y \in B$$

and at this point just use the gaussian upper bound for the heat kernel $p(t,x,y)$ of the whole manifold $M$, e.g. the famous Li-Yau estimate [3, Thm. 3.2]. Here one should be careful though as it seems that you are asking for a precise gaussian factor $\exp(-d^2(x,y)/4t)$ which is in general not possible to achieve globally, as these estimate will fail at the cut locus, but of course one can prove global estimates with a gaussian factor, such as

$$ p(t,x,y) \leq \frac{C}{t^{n/2}} e^{-\frac{d^2(x,y)}{(4+\epsilon)t}} $$

for some $\epsilon >0$, cf. the aforementioned Li-Yau reference.

For the lower bound, if e.g. $x$ is a point at the boundary of your ball, then $p_B(t,x,\cdot)=0$, so you cannot have a lower bound as you ask. However, if we fix the central point of the ball, that there is a classical results by Debiard-Gaveau-Mazet and Cheeger-Yau which might be what you need. It says that if $o\in M$, $B=B_R(o)$ and $k$ is a lower bound for the Ricci curvature on $B$, that is $\mathrm{Ric}(B)\geq k(n-1)$, then

$$
p_B(t,o,y) \geq Q(t,d(o,y))
$$

where $Q$ denotes the heat heat kernel on a ball of radius R of the simply connected n-dimensional manifold with constant curvature equal to k. You can find a statement and references in [2]

[1] *Grigor’yan, Alexander*, Heat kernel and analysis on manifolds, AMS/IP Studies in Advanced Mathematics 47. Providence, RI: American Mathematical Society (AMS); Somerville, MA: International Press (ISBN 978-0-8218-4935-4/hbk). xvii, 482 p. (2009). ZBL1206.58008.

[2] *Chavel, Isaac*, Eigenvalues in Riemannian geometry. With a chapter by Burton Randol. With an appendix by Jozef Dodziuk, Pure and Applied Mathematics, 115. Orlando etc.: Academic Press, Inc. XIV, 362 p. {$} 62.00; \textsterling 46.50 (1984). ZBL0551.53001.

[3] *Li, Peter; Yau, Shing Tung*, **On the parabolic kernel of the Schrödinger operator**, Acta Math. 156, 154-201 (1986). ZBL0611.58045.

false. In the case of the sphere, a lovely articleSharp estimates of the spherical heat kernelby A. Nowak, P. Sjögren and T.Z. Szarek, DOI:10.1016/j.matpur.2018.10.002 gives an exact bound of the heat kernel which is different from what is written in the question. $\endgroup$