Congratulations to the two student members who sent correct answers to this puzzle: Yudong Chen (University of Cambridge, UK) and Zhen Huang (Columbia University, USA). Here’s Anirban DasGupta’s solution:

Using the notation of the problem, the recorded values $Y_1,Y_2,\cdots$ are iid with
$E(Y_i)=\sum _{i=1}^{\mathrm{\infty }}(ic[F((i+1/2)c)-F((i-1/2)c)])$,
where $c=\frac{1}{100}$, and $F(x)=1-e^{-x/\lambda }$ is the CDF of the exponential distribution with mean $\lambda$. On using the formula
$\sum _{i=1}^{\mathrm{\infty }}i{a}^i=\frac{a}{(1-a{)}^2}$ for $|a|<1$, we get $\overline{Y}\stackrel{P}{\to }E(Y_i)=\frac{e^{-\frac{1}{200\lambda }}}{100(1-e^{-\frac{1}{100\lambda }})}=\lambda -\frac{1}{24\times {10}^4\lambda }+\frac{7}{576\times {10}^9{\lambda }^3}+O({\lambda }^{-5})$ as $\lambda \to \mathrm{\infty }$. Thus, even for large $\lambda$, there is a small persistent bias in the estimator that we actually use in real life, and the real life estimator is inconsistent.