Inflation results from the possibility of debt monetisation

Private sector companies are not able to cause inflation. We only get inflation, other than by directly printing money, if the Government has issued liabilities which, if defaulted on will result in the Government issuing more money to pay back the debt. Bank deposits are a liability of the bank that are guaranteed by the Government and hence taxpayer. It is because monetisation of the debt will (or might) result that issuing new liabilities causes inflation. If it is imagined that the credit of a non-bank private company might result in new money being issued to repay it, then that might result in inflation.

If a company with close ties to the Government issued debt, the Government might deem it in the 'National interest' to redeem the debt with new money to keep the company afloat, then this company would likely cause inflation by issuing debt. A company that is too big to fail will cause inflation by issuing excess liabilities, in a fiat regime. Also Government liabilities such as healthcare promises, or public sector pensions will result in inflation if people consider them valuable in spite of the Government not having the means, without printing, to pay them off. If we expect the Government not to default on its obligations, then any new obligation will affect the price of money...

Any claims on money (in the future) result in inflation if we consider that there is a chance that the debt will be repaid, with new money. Any promise that might be monetised will result in inflation. State liabilities are a, in a sense, a promise to print in the future and can result in inflation.