Assume you have a filtered probability space $(\Omega,\mathcal{F},P,\mathcal{F}_t)$. Lets say you have the financial market with the bank process:
$$dR_0(t,\omega)=\rho(\omega,t)R_0(t,\omega)dt,$$
where $\rho$ is a progressively measurable process and it is assumed that $P(\int_0^T|\rho(\omega,t)R(\omega,t)|dt<\infty)=1$. And you have a risky asset which is modelled by
$$dR_1(\omega,t)=\mu(\omega,t)dt+\sigma(\omega,t)dB_t,$$
where $\mu, \sigma$ are progressively measurable and $P(\int_0^T|\mu(\omega,t)|dt<\infty)=1$ and $P(\int_0^T|\sigma(\omega,t)|^2dt<\infty)=1$.
Now assume that $\sigma(\omega,t)=0$. Then the risky asset is just modelled by
$$dR_1(\omega,t)=\mu(\omega,t)dt.$$
Can we show that we have arbitrage in this case?
The reason I suspect this to be true is that if we we instead had a time-discrete market with $t_0=0<t_1<t_2<\cdots<t_n=T$ and we modelled it by
$$\Delta R_1(\omega,t_i)=R_1(\omega,t_{i+1})-R_1(\omega,t_{i})=\mu(\omega,t_i)\cdot \Delta t_t,$$
then we would have had arbitrage. Because since $\mu$ is adapted, it is known at time $t_i$ hence at time $t_i$ we would know what the value of the risky asset would be at time $t_{i+1}$. We would also know what the value of the bank process would be at time $t_{i+1}$ at time $t_i$. So we could either borrow money and buy the stock or short the stock and put money in the bank, depending on what gave highest return. However, can we show that it will lead to arbitrage in the time-continuous case as well?
Update
I think I have a solution. It goes like this:
Look at the dsicounted process $R_1/R_0$, this is an Itö-process by looking at the function $F(x,y)=x/y$ and noting that $R_1/R_0=F(R_1,R_0)$. F satisfies the hypothesis in the Itö-formula since $R_0\ne 0$. Since $dR_1$ and $dR_0$ don't have any $dB$ terms it follows from Itös formula that $dF(R_1,R_0)$ don't have any $dB$ terms. Hence
$$\frac{R_1(t)}{R_0(t)}=\frac{R_1(0)}{R_0(0)}+\int_0^tK(\omega,s)ds,$$
where we find $K$ from Itö's formula. Since Itös formula tells us that $R_1/R_0$ is an Itö-process we have that
$$P\left(\int_0^T|K(\omega,t)|dt<\infty\right)=1.$$
The fundamental theorem of asset-pricing tells us that there is no arbitrage if and only if there exist an equivalent local martingale measure $Q$ where $R_1/R_0$ is a local martingale. Assume for contradiction that $Q$ exists.
Now we have that $$R_1/R_0$$ is defined $\omega$-wise $P$-a.s. And since $Q$ is equivalent to $P$ we must also have
$$Q\left(\int_0^T|K(\omega,t)|dt<\infty\right)=1.$$
Now look at $K$, $\omega$-wise. Then it is an $L^1[0,T]$-function. And it is known that $F(s)=\int_0^t f(s)ds$ has bounded variation for $f \in L^1[0,T]$.
Hence under $Q$ we have that $R_1/R_0$ is a continuous local martingale with finite variation. This means that it is constant. Hence
$$Q(\{\omega: R_1(\omega,\cdot)/R_0(\omega,\cdot)=\text{Constant}\})=1.$$
But since $P$ and $Q$ are equivalent we have
$$P(\{\omega: R_1(\omega,\cdot)/R_0(\omega,\cdot)=\text{Constant}\})=1.$$
This means that
$$R_1(\omega,t)=R_0(\omega,t)\cdot M(\omega).$$
But this also holds for $t=0$ hence
$$R_1(\omega,t)=\frac{R_1(0)}{R_0(0)}\cdot R_0(\omega,t),$$
Hence we are limited to the trivial case where the risky asset is a multiple of the bank-process.
Intuitively it is quite easy to see that there is arbitrage as long as the drifts are unequal with positive probability. Here you would invest in the asset with higher drift and sell the asset with lower drift.
More formally you could use the result that the model is free of arbitrage if and only if there exists a risk neutral measure under which the stock price discounted with the bond is a martingale.
Let $\tilde{\mu}(\omega,t)=\frac{\mu(\omega,t)}{R_1(\omega,t)}$ and $\tilde{\sigma}(\omega,t)=\frac{\sigma(\omega,t)}{R_1(\omega,t)}$.
If $\tilde{\mu}(\omega,t) = \rho(\omega,t)$ the physical and risk neutral measures coincide. However, if not under this measure $B_t$ evolves as
$$dB_t^{Q}=\frac{\tilde{\mu}(\omega,t)-\rho(\omega,t)}{\tilde{\sigma}(\omega,t)}+dB_t$$
(see any book on math finance). Now if $\sigma(\omega,t)=0$ this risk neutral measure does not exist. There is no measure under which the discounted stock price were a martingale, hence there is arbitrage.