In short: lockdown is not causing an increase in suicide rates - in fact deaths from all causes are down - but we do have a problem.
The coroner recently released their report on suicide rates for the past year. The media release can be found at this link: https://coronialservices.justice.govt.nz/assets/Documents/Publications/Chief-Coroner-Suicide-Stats-2020-Media-Release.pdf, the full report at this one: https://coronialservices.justice.govt.nz/assets/Documents/Publications/2020-Annual-Provisional-Suicide-Statistics.pdf
I commend to you the full report. In spite of the current years' figures being a decrease on the prior year, the trend-line remains upwards over several years. Ethnic breakdowns of the rates are worth reviewing. Maori rates of suicide are highest, but this is followed by European rates, then Asian, then Pacific people. Male rates of suicide are 2.5 to 3 times the rates for females.
To provide context, the chart below (with thanks to Ourworldindata.org) shows the rate per 100,000 of population, from 2000 to 2016, with some selected countries and groups of countries for comparison. It is the comparison with the UK that strikes me in particular. The chart is interactive. Feel free to change countries and explore.