Can we find a complete connected noncompact Riemannian manifold $(M^n,g)$ such that the curvature operator $Rm>0$ and 
$$
\inf\_{p \in M} \text{Vol}\_gB(p,1)=0?
$$



Positively curved manifold with collapsing unit balls
The answer is negative if $\dim M=2$ and positive otherwise, as shown in the paper:


Croke, C. B., & Karcher, H. (1988). VOLUMES OF SMALL BALLS ON OPEN MANIFOLDS: LOWER BOUNDS AND EXAMPLES. AMERICAN MATHEMATICAL SOCIETY (Vol. 309). <https://www.ams.org/journals/tran/1988-309-02/S0002-9947-1988-0961611-7/S0002-9947-1988-0961611-7.pdf>


The negative result is an immediate consequence of Theorem A, while the positive one follows from Example 1. Their construction for the latter is as follows:


Consider the $d$-dimensional hypersurface given by the paraboloid $\mathcal P = \{x\in \mathbb R^{d+1}\mid x\_{d+1}=x\_1^2+\ldots+x\_d^2\}$. At any point $A$ of $\mathcal P$ one can consider its tangential cone $\mathcal C$, i.e., the Euclidean cone of vertex $V$ with axis along the line $AV$ and that intersects $\mathcal P$ tangentially. One then can consider $\tilde {\mathcal P}$ to be $\mathcal P$ with $\mathcal C$ "attached" (see the Figure at p.765). 


This is a piecewise $C^\infty$ hypersurface, with a conical singularity at $V$ and that is $C^1$ at the intersection $\mathcal P\cap \mathcal C$. Outside of these regions, the curvature of $\tilde {\mathcal P}$ is $K>0$, since the curvature of $\mathcal P$ is positive and the one of the cone is zero. Moreover, it is always possible to smooth a conical singularity preserving the $K>0$ bound, and the same is true for the $\mathcal P\cap \mathcal C$ part. (You can maybe look at this [thesis](https://d-nb.info/1054638039/34#page13), e.g., Lemma 3.2.4.) So we have constructed a smooth hypersurface with $K>0$, and where we can estimate the volume of balls contained in the smoothed conical region by the volumes of the corresponding regions in the non-smoothed cones. Moreover, this operation can be done with a sequence of tangent cones $\mathcal C\_n$, as soon as they are sufficiently distant one from the other.


The estimation of the volumes of the balls on the tangent cones is the object of the claim at the end of p.765. In particular, there the authors show that if the vertex of the cone is $V = (k+1,\ldots, 2k+1)$ for $k\ge 5$ (I'm fixing $r=1$), this volume is bounded by the volume of a "spherical spindle" :
$$
\operatorname{vol}(B(V,1))\le C\_d\sin^{d-2}\theta ,
$$
where $\theta$ is an angle bounded by $\tan^2\theta\le 6/k$. (Here I chose $\varepsilon =1\le k^2/(2k+2)$.)


This shows that it is possible to attach to $\mathcal P$ a sequence of cones $\mathcal C\_n$ with vertices $V\_n\to +\infty$, and smooth them out so that
$$
\lim\_{n\to +\infty}\operatorname{vol}(B(V\_n,1))\le (C\_d+1) \lim\_{n\to +\infty} \sin^{d-2}\left(\theta\_n\right) = 0.
$$
This proves that the obtained hypersurface (which has positive curvature) satisfies your requirement.


